fl STUDY ON
FLDffi FLOW, HEAT TRANSFER, MORPHOLOGY AND
MACROSEGREGATION IN CONTINUOUS CASTING OF STEEL
by
SHIV KUMAR CHOUDHARY
i
1 ^ ^ 1
I
WAAiiim m MAtSfOMs Jtm mitAismcmbL engineering
INDIAN INSTITUtE OF TECHNOLOGY, KANPUR
i STTOY ON
FLUID FLOW, HEiT TRJINSFER, MORPHOLOGY AND
MACROSEGREGATION IN CONTINUOUS CASTING OF STEEL
A Thesis Submitted
in Partial Fulfilment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
by
SHIV KUMAR CHOUDHARY
CO the
DEPARTMENT OF MATERIALS AND METALLURGICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, KANPUR
September, 1993
672 . - 7 ^-
“ 7 JUN 1994
CENTRAL L'?RAR7
I t ^
xiisiis
P^a^-l ffZ'D -cfre _ S 71 ,)
I
I
f
DEDICATED
MY FATHER
\ r' « r ‘
CERTFICATE
It IS certified that the work contained in the thesis
entitled "A STUDY ON FLUID FLOW, HEAT TRANSFER, MORPHOLOGY AND
MACROSaEGREGATION IN CONTINUOUS CASTING OF STEEL" by Shiv Ktimar
Choudhary has been carried out under our supervision and that this
work has not been submitted elsewhere for a degree.
^(A. Ghosh)
Professor
Department of Metallurgical Engineering
Indian Institute of Technology
Associate Professor
Kanpur
ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation and
gratitude to Prof. A. Ghosh and Dr.D. Mazuindar for their able
guidance, valuable suggestions and patience during the course of
this study.
The financial support received for this study from the
National Mission on Iron and Steel, Ministry of Iron and Steel,
Government of India, is gratefully acknowledged. The special
thanks are due to Tata Steel, Jamshedpur for providing the billet
samples for the present study. The help received from Mr.
Rameshwar Sharma of Research and Development Division, Tata Steel
is also gratefully acknowledged. The author sincerely appreciates
the help rendered by the National Metallurgical Laboratory in the
chemical analysis of samples.
The author is grateful to Prof. T. Sundarajan for the
discussions he had with him during the different stages of the
present work. He is also thankful to Prof. A.K. Biswas, Prof.
R. K. Ray, Prof. Brahma Deo and Prof. N. Chakraborti for their
constant encouragement and help during this work.
The author is also grateful to Mr. T.K. Roy and Mr. A.
Sharma for their constant help and cooperation throughout the
study. Special thanks are also due to all his friends, specially
S. Ghosh, G.G. Roy, P.V.K. Reddy, S.N. Singh, N.K. Nath, K.K.
Singh, S.K. Shrivastava, Dr- D. Bandyopadhyay and Dr. S.K. Dutta
t
for their assistance in completing the work. The timely help
rendered by Mr. K. Rao is greatly appreciated.
The author sincerely acknowledges the help of Mr. K.P. ;
Mukherjee in the photography work, Mr. B. D. Biswas for the |
painstaking effort in typing the manuscript and Mr. V.P. Gupta for
the nice tracing of figures. Thanks are also due to Mr.R.C.Sharma
and Mr.V.P.Vohra of Metallurgical Engineering Workshop for their
assistance in the fabrication, and other jobs.
The goodwill and support of many friends and well wishers
was a great asset to the author during his stay at the I.I.T.
Kanpur; it has not been possible to present the large list of
their names. The assistance and support from the Department of
Materials and Metallurgical Engineering, Computer Centre, Glass
Blowing Shop, and Academic Section of I.I.T. Kanpur are
thankfully acknowledged.
His friend Chidanand showered affection and vital friendship
during the progress of this work. His father and family members
constantly encouraged and assisted him in various ways for which
he records his indebtedness. His wife Ran j ana supported him
through remarkable and patient understanding.
LIST OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
LIST OF SYMBOLS
SYNOPSIS
CHAPTER
1 INTRODUCTION
1.1 DESCRIPTION OF CONTINUOUS CASTING OF STEEL
1.2 DEVELOPMENT OF CONTINUOUS CASTING OF STEEL
1.3 OBJECTIVE OF THE PRESENT STUDY
1.3.1 Heat Transfer and Solidification
During Continuous Casting of Steel
1.3. 1.1 Plan of work for mathematical
modelling of heat transfer
1.3.2 Macrosegregation and Morphology
in Continuously Cast Products
1.3. 2.1 Plan of work on macrosegregation
and morphology
1.4 PRESENTATION OF CHAPTERS IN THE THESIS
2 MATHEMATICAL MODELLING OF CONTINUOUS CASTING
OF STEEL VIA ARTIFICIAL EFFECTIVE THERMAL
CONDUCTIVITY APPROACH
2 . 1 INTRODUCTION
2.2 LITERATURE REVIEW
2.3 FORMULATION OF THE GOVERNING EQUATION
FOR THE PRESENT STUDY
2.3.1 Assumptions in Modelling
2.3.2 Governing Heat Flow Equation
2.3.3 Modelling of Axial Heat Conduction
Term in the Governing Equation
Page
xi
xvii
xviii
xxiii
1
2
6
9
10
12
13
14
15
16
16
18
28
28
30
32
Vi
CHAPTER Page
2.3.4 Modelling of Latent Heat Release 33
Effect
2.3.5 Boundary Conditions 36
2.4 NtJMERICAL SOLUTION 40
2.4.1 Numerical Solution Procedure 40
2.4.2 The Computer Program ‘ 52
2.5 RESULTS AND DISCUSSIONS 55
2.5.1 Sensitivity of Computation to 55
the Choice of Grid Distribution
2.5.2 Influence of Various Numerical 58
Approximations on the Computed
Results
2. 5. 2.1 Arithmetic Mean vs. Harmonic Mean 58
Approximation for Estimating the
Control Volume Face Thermal
Conductivity
2. 5. 2. 2 Lower order vs. higher order 62
interpolations for estimating
the cast surface temperatures
2. 5. 2. 3 Influence of different numerical 66
integration procedure for the mould
heat flux expression
2.5.3 Influence of Axial Conduction on 70
the Computed Results
2.5.4 Influence of Modelling Procedures 73
Applied to Approximate Heat Conduction
in the Mushy zone
2.5.5 Influence of Mould Heat Flux 76
on the Computed Results
2. 5. 5.1 Instantaneous vs. average mould heat 76
flux expressions as the surface
boundary condition in the mould region
2. 5. 5. 2 Confidence limit of mould heat flux 80
expression and its likely influence on
the accuracy of computed results
vii
HAPTER Page
2.5.6 Sensitivity of Computation to the 80
Choice of Effective Thermal
Conductivity Values
2.5.7 Comparison of Results with Literature 85
Experimental Data
2.6 SUMMARY AND CONCLUSIONS 91
3 MATHEMATICAL MODELLING OF HEAT TRANSFER IN 95
CONTINUOUS CASTING OF STEEL VIA CONJUGATE
FLUID FLOW AND HEAT TRANSFER APPROACH
3 . 1 INTRODUCTION 95
3.2 LITERATURE REVIEW 96
3.3 FORMULATION OF TRANSPORT EQUATIONS FOR THE 103
PRESENT STUDY
3.3.1 Assumptions in Modelling 103
3.3.2 Governing Equation of Fluid Flow 104
Within the Liquid Pool and
Boundary Conditions
3.3.3 Modelling of Turbulence Within llO
- the Liquid Pool
3.3.4 Governing Equation of Heat Flow 113
and Boundary Conditions
3.3.5 Non-dimensionalization of 116
Governing Equations
3.3.6 Modelling of Fluid Flow in the 120
Mushy Zone
3.3.7 Choice of the Outflow Boundary 121
3.4 NUMERICAL SOLUTION OF THE GOVERNING PARTIAL 122
DIFFERENTIAL EQUATIONS
3.4.1 Numerical Solution Procedure 122
3.4.2 Numerical Procedure for Incorporating 127
the Influence of Solidifying Shell on
Fluid Flow and Heat Transfer
3.4.3 The Computer Program 131
r \
viii
HAPTER Page
3.5 RESULTS AND DISCUSSION 135
3.5.1 Some Considerations on the Scope of 135
convergence of a Multidimensional
Coupled Fluid Flow Heat Transfer
Problem
3.5.2 Sensitivity of Computation to the 139
Choice of Effective Viscosity Value
3.5.3 Modelling of Flow in the Mushy Zone 141
and Its Influence on the Computed
Results
3.5.4 Influence of Thermal Buoyancy Force 144
on the Computed Results
3.5.5 Role of Prescribed Temperature vs. 146
Insulated Surface, Out Side the
Pouring Stream, as Meniscus Boundary
Conditions
3.5.6 Predicted Flow Field Within the 148
Liquid Pool of Solidifying Casting
3.5.7 Comparison of Numerical Predictions 154
with Reported Experimental Measurements
3.6 SUMMARY AND CONCLUSIONS 162
4 STUDY ON MORPHOLOGY AND MACROSEGREGATION 167
IN CONTINUOUSLY CAST STEEL BILLETS
4 . 1 INTRODUCTION 167
4.2 LITERATURE REVIEW 169
4.2.1 Influence of Morphology of Cast 178
Structure on Macrosegregation
4.2.2 Influence of Superheat 181
4.2.3 Influence of Electromagnetic Stirring 181
4.2.4 Role of Peritectic Transformation 184
4.2.5 Fluid Flow, Bulging and Centreline 189
Segregation
4.2.6 Measures to Reduce centreline 191
Segregation
ix
CHAPTER Page
4.2.7 Problems of Quantitative Measurement 194
of Macrosegregation
4.2.8 Macrosegregation and New Measurement 199
Techniques
4.3 EXPERIMENTAL PROCEDURE 201
4.3.1 Plant Data and Sample Collection 201
4.3.2 Macroetching of Transverse Section 206
of Billets
4.3.3 Chemical Analyses of Samples 207
4.4 RESULTS AND DISCUSSIONS 210
4.4.1 Results and Discussions on 211
Macrostructural Examination
4. 4. 1.1 Measurement of equiaxed zone size 211
4. 4. 1.2 Influence of tundish superheat on 217
equiaxed zone size
4.4.2 Results and Discussions on 221
Macrosegregation Studies
4. 4. 2.1 Results 221
4. 4. 2. 2 Comparison of segregation levels at 223
centreline and at columnar-equiaxed
transition (CET) boundary
4. 4. 2. 3 Quantitative relationship between 227
r„ and r„
s c
4. 4. 2. 4 Correlation between r_ and r with 231
s o
the help of segregation ec[uations
4. 4. 2. 5 Relationship of r and r at CET 238
s c#
boundary with fractional solidi-
fication
4.5 CORRELATION OF MACROSEGREGATION AND MORPHOLOGY 244
DATA WITH PREDICTION OF HEAT TRANSFER MODEL
4.5.1 Estimation of Correct Pouring 245
Temperature of Liquid Steel
4.5.2 Comparison of Measured Location of 249
CET Boundaries with Those Predicted
from Mathematical Model
X
CHAPTER
4.6 SUMMARY AND CONCLUSIONS
5 SUMMARY AND CONCLUSIONS
5.1 MATHEMATICAL MODELLING BY ARTIFICIAL
EFFECTIVE THERMAL CONDUCTIVITY APPROACH
5.2 MATHEMATICAL MODELLING BY CONJUGATE FLUID
FLOW - HEAT TRANSFER APPROACH
5.3 STUDY ON MACROSEGREGATION AND MORPHOLOGY
5.4 CORRELATION OF MACROSEGREGATION AND
MORPHOLOGICAL STUDIES WITH HEAT TRANSFER
MODELLING
5.5 SUGGESTIONS FOR FURTHER WORK
REFERENCES
Page
253
256
256
258
260
261
262
264
LIST OF FIGURES
Figure
Title
Page
1.1
1.2
1.3
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Schematic of typical slab casting machine
Schematic of three zones of heat extraction
during continuous casting of steel
Crude steel and CC production worldwide
Schematic of a typical continuous casting operation
and a three dimensional volume element in the
casting strand illustrating the concept of energy
balance applied to derive Eq.2.1
Relevant section of the idealized iron-carbon
equilibrium diagram
Schematic of the calculation domain in two
dimension and the associated boundary
conditions applied to solve Eq.2.1
Schematic of grid distribution in (a) one quarter
of a square billet (b) in the central vertical
plane and (c) in a transverse plane
A typical three dimensional control volume in
cartesian coordinate system
Typical boundary control volumes in a 2D
calculation domain
Flow chart of computer program for the model based
on effective thermal conductivity concept
Variation of shell thickness with distance below
meniscus for different grid configuration
(Round billet , dia. = 0.115 m)
Variation of surface temperature with distance
below meniscus for different grid configurations
(Round billet^^, dia. = 0.115 m)
Effect of arithmetic mean and harmonic mean
approximation techniques (e.g. for control volume
face conductivity) on predicted shell thickness in
. 22
a square billet
3
5
8
31
35
37
41
42
49
54
56
57
60
xli
Figure
Title
Page
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
Effect of arithmetic mean and harmonic mean
approximation (e.g. for control volume face
conductivity) on predicted midface temperature in a
square billet^^
Boundary control volumes considered for (a) lower
order and (b) higher order interpolation methods
for estimating cast cast surface temperature
Effect of lower order and higher order
interpolation techniques on predicted
midface temperature in a square billet
A 2D representation of a typical boundary control
volume in the central vertical plane of a square
billet
Effect of different integration routes applied to
the mould heat flux expression on predicted shell
thickness in a square billet caster
Influence of axial conduction term in the governing
heat flow equation on predicted shell thickness
Influence of axial conduction term in the governing
heat flow equation on predicted surface temperature
Influence of mushy zone treatment on predicted
shell thickness
Influence of mushy zone treatment on predicted
midface temperature of a square billet
Effect of instantaneous and average mould heat flux
as boundary condition at the mould wall on
predicted shell thickness
Influence of variation in mould heat flux on
predicted shell thickness
Influence of different effective thermal
conductivity values on predicted shell thickness of
27
a typical slab caster
Present estimates of solid shell thickness in a
billet caster for different effective thermal
conductivity values and their comparison with
22
experimental measurements
61
63
65
67
69
71
72
74
75
78
79
82
83
xlii
Figure
2.24
2.25
2.26
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
Title
Comparison between predicted and experimental shell
thickness of a typical square billet caster^ ^
(Billet : 0.14 x 0.14 m)
Comparison between predicted and experimental shell
thickness of a typical square billet caster^ ^
(Billet : 0.133 x 0.133 m)
Comparison between predicted and experimental shell
thickness of a typical round billet caster^^
(Billet dia. : 0.115 m)
Schematic of the flow pattern in the liquid pool of
a continuously cast billet
Schematic representation of the calculation domain
and the boundary conditions applied in the
computation of velocity and temperature fields
Schematic of the grid layout and control volumes
for vector (u & v) and scalar (p & T) variables
Schematic of three typical control volumes for
scalar (i.e. p & T) and vector (u & v) variables
employed in the numerical computation scheme
Schematic of typical radial velocity control volume
located in the vicinity of the solidification front
and evaluation of blockage ratios for various
control volume faces
Flow chart of the model applied to the numerical
computation of velocity and temperature fields
in CC
Change in dimensionless axial velocity component at
the monitoring location (i.e. node (6,5)) with the
progress of iterations
Change in dimensionless temperature at the
monitoring location with the progress of iteration
Influence of average effective viscosity value in
the fluid flow equations on estimated shell
thickness
Influence of mushy zone viscosity value on
the predicted shell profile
Page
86
87
89
106
109
124
125
130
134
137
138
140
143
xiv
Figure
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
Title
Influence of buoyancy force term in the momentum
balance equation on the predicted shell profile
Influence of two different types of meniscus
boundary conditions (applied to the temperature
equation ) on the predicted shell profile
Schematic representation of the flow field with (a)
radial flow nozzle and (b) straight bore nozzle
Computed two dimensional flow field in a typical
round billet
Computed flow field in the central vertical plane
of a typical square billet section
Computed flow field in the central vertical plane
. 22
of a typical square billet section
Comparison between the computed shell thickness and
the corresponding best fit data for a typical round
billet^ ^caster
Comparison between the present estimate of the
shell profile and the corresponding experimental
measurement of a typical round billet caster
Comparison between the present estimate of the
shell profile and the corresponding experimental
measurement of a typical square billet caster
(Billet size = 0.14 x 0.14 m sq.)
Comparison between the present estimate of the
shell profile and the corresponding experimental
22
measurement of a typical square billet caster
(Billet size = 0.133 x 0.133 m sq.)
Comparison between the temperature profiles
predicted by the conjugate fluid flow-heat
transfer model and the effective thermal
conductivity model at the mould exit of a
round billet^^ caster
Typical concentration profile as observed in CC slab
Macrostructure of a low carbon steel billet
Axial segregation index as a function of equiaxed
zone size
Page
145
147
149
151
152
153
155
156
157
158
160
4.1
4.2
4.3
168
179
182
XV
Figure Title Page
4.4 Influence of carbon content of steel on columnar zone 186
length (a) CC billet; Samarasekera et al^® (b) 8620
steel ingots; Hurtuk and Tzavaras^^
4.5 Formation of mini-ingot in continuous casting 190
4.6 Influence of bulging on centreline segregation 192
4.7 Interdendritic fluid flow in continuous casting 193
(a) limiting case, all flow vertical - no
segregation results;
(b) flow resulting in negative segregation at
cast centre;
(c) flow resulting in positive segregation
4.8 Some features of macrosegregation in longitudinal 196
section of CC products (schematic)
4.9 Segregation profiles of carbon and sulphur along 197
the centreline of a typical steel billet
4.10 Photograph of a drill surface 209
4.11 Photographs of macroetched surface of billet 215
samples with eguiaxed zones as follows:
(a) symmetric (type I)
(b) asymmetric about one axis (type II)
(c) asymmetric about both axes (type III)
4.12 Sketches of columnar-equiaxed transition 216
boundaries for photographs in Fig. 4. 11
(a) symmetric
(b) asymmetric about one axis, and
(c) asymmetric about both axes
4.13 Influence of tundish superheat on the percentage 220
equiaxed zone area in billet samples
4.14 Relationship between r at the centreline and 225
columnar-equiaxed transition (CET) boundary
4.15 Relationship between r at the centreline and CET 226
boundary
4.16 Relationship between r„ and r at the centreline in 228
s c
billet samples
Relationship between r and r at the CET boundary
in billet samples
4.17
230
xvl
Figure Title Page
4.18 Testing of applicability of equilibrium solidification 233
model to segregation data at GET boundary
4.19 lnr_/lnr_ values of different samples for the 236
centreline and the GET boundary
4.20 Variation of Inr with ln(l - f„) for GET boundary 240
s s
4.21 Variation of Inr with ln(l - f^) for GET boundary 241
w S
4.22 Influence of bulk liquid flow (v) on the rate of 243
solidification (R) and concentration profiles of
different solute elements; as observed by Takahashi
*76
et al in a controlled laboratory experiment.
4.23 Relationship between superheat of molten steel in 247
the tundish and that in the mold^^.
4.24 Gomparison between measured and computed distance 251
of GET boundary from centre of the billet samples
with uncorrected pouring temperature.
4.25 Gomparison between measured and computed distance of 252
GET boundary from centre of the billet samples with
. corrected pouring temperature.
LIST OF TABLES
Table Title Page
2.1 Numerical data of CC used in the present computation 94
2.2 Thermophysical properties of steel used in computation 94
3.1 Casting conditions considered for numerical simulation 165
3.2 Thennophysical properties of steel used in the present 166
numerical computations
4.1 Values of for solidification of iron 171
4.2; Characteristics of continuous casting machine at TATA 205
STEEL
4.3: Data on billet samples collected from Tata Steel 212
4.4; Measured area percent of various structure in 214
transverse section of CC billets
4.5; Estimated liquidus temperature of billet samples 219
using different correlations
4.6; Analyses of carbon and sulphur at centreline and 222
CET boundary of billet samples
4.7; Segregation ratios of carbon and sulphur at the 224
centreline and CET boundaries of different billet
samples
4.8; Results of correlation between measured CET 250
boundary and those predicted by the
mathematical model
LIST OF SYMBOLS
Symbol
a
Ap, Ag, A^, Ajj, Ag, Ag, A^
^E^new
^Eq
b
BRe^ BR^, BR^
C
^0
c
s,o
C
C,L
: Billet size in the direction
of X-axis (m)
: Coefficients of the
discretization equation
: Modified coefficients of discre-
tization equation after blockage
ratio correction
: Percentage of equiaxed zone area
I Billet size in the direction of
Y-axis (m)
: Blockage ratios of various faces
of velocity control volume
: Specific heat of steel (J kg"^ °C)
: Initial concentration of solute in
the liquid (wt. pet.)
: Nominal carbon concentration
of steel (wt. pet)
I concentration of solute i at the loca-
tion under consideration (wt. pet)
: Nominal concentration of
of solute i (wt. pet)
: Equilibrium solute concentration
in the liquid phase (wt. pet)
: Equilibrium solute concentration
in the solid phase (wt. pet)
: Nominal sulphur concentration
of steel (wt. pet)
: Equilibrium carbon concentration of
solid phase in the mushy zone (wt. pet)
: Equilibrium carbon concentration of
liquid phase in the mushy zone
(wt. pet)
; Diameter of the mould (m)
: solid fraction in the mushy zone
g
Gr
h_
eff
s
0
m
K
K
eff
^eff,e
etc.
"t
[i
h
5
le etc.
hsL
Is
xix
: Solid fraction at the top and
bottom faces of a control volume
respectively
—2
Acceleration due to gravity (ms )
: Grashoff ntimber
: Spray heat transfer
coefficient (W m~^
: Equilibrium partition coefficient
: Equilibrium partition coefficient of
carbon
: Effective partition coefficient
: Equilibrium partition coefficient of
sulphur
; Mass transfer coefficient (m s~^)
: Molecular thermal conductivity
of steel (W m”^
: Artificial effective thermal
conductivity of liquid (W m”^
: Effective thermal conductivity
value at one face of a control
volume (W m”^
: The turbulent thermal
conductivity (W m**^
; Caster/domain length
simulated (m)
; Mould length (m)
: Secondary cooling zone length (m)
: The mass flow rate of liquid
steel (kg s~^
-2
; Pressure (N m )
: The Peclet Number
: Heat flux at one face of
—2
a control volume (W m )
-2
: Instantaneous mould heat flux (W m )
-2
: Average mould heat flux (W m )
-2
: Heat flux at the cast surface (W m )
XX
R
Re
r
S
Sp
Su
Su
T
t
'^liq
’^sol
m m m iTi m rp rn
P' E' W' N' S' B' T
T etc.
e
: Heat fluxes along X,Y, and Z
coordinates (W m )
: Slze/radius of billet (m)
: Reynolds Number
: Radial distance (m)
: Pouring stream radius (m)
: Degree of carbon segregation
: Degree of sulphur segregation
: The source term in the
discretization equation
: The slope of the linearized
source term
: Constant part of the
linearized source term
: Source term in the axial direction
—3
momentum balance equation (N m )
: Source term in the radial/ transverse
direction momentum balance
equation (N m )
: The temperature variable (°C)
Time (s)
: Liquidus temperature of steel (°C)
: Dwell time of casting in the
mould (s)
: Solidus temperature of steel (°C)
: Temperature at the various nodal
points (°C)
: Temperature at one face of a
control volume (°C)
: Ambient temperature (°C)
: The pouring/casting temperature (°C)
: The cast surface temperature (°C)
: Spray water temperature (°C)
: Axial velocity component variable (ms'
: Velocity of the pouring stream at
the entrance of the mould (m s”^)
: Casting speed (m s”^)
v
: Radial/ transverse velocity
Y
Z
Greek letters
AT
AHf
AX, AY, AZ
a
a
^L'^S
P
c
0
<p
h, ^N' ’^s
\
xxj
component variable (m s ■*■)
: One of the transverse coordinate
: Position of the solidification
front from the axis of symmetry (m)
: Another transverse coordinate
; The axial coordinate
: Super heat of liquid steel or
Temperature difference (°C)
; Latent heat of solidification of
steel (J kg”^)
: Length of control volume faces in
three mutually perpendicular
coordinate directions (m)
: Dendrite arm spacing
: Index of coordinate system
: The slope of liquidus and solidus
lines respectively in the
iron-carbon equilibrium diagram
: Distance between the various nodes
in three mutually perpendicular
coordinate axes
: Intercepts of the liquidus and
solidus lines respectively in the
iron-carbon equilibria diagram
; Coefficient of volumetric expansion
of liquid steel
: Emissivity of the oxidized iron
surface (= 0.85)
: Rate of temperature change (°C s
: General dependent variable
(i.e. u, V, p or, T) of the
discretization equation
: Values of at the various nodes in
the general discretization equation
: Volume fraction of liquid in the
mushy zone
xxii
r
eff
X
M
^eff
0
®o
P
Pl
Ps
C
Superscript
: Effective thermal conductivity
derived from the turbulence
model (W m"^ °c”^)
: Index of coordinate dimension
(i.e. a l-D, 2-D or, 3-D problem)
: Molecular viscosity of liquid
steel (kg m”^ s”^)
; Effective viscosity (kg m s )
: Turbulent viscosity (kg m”^ s”^)
2 —1
: Turbulent kinematic viscosity (m s )
: Temperature in absolute scale (°K)
; Pouring temperature in absolute
scale (°K)
; Cast surface temperature in
absolute scale (°K)
: Ambient temperate in absolute
scale (°K)
—3
: Density of steel (kg m )
—3
; Density of liquid phase (kg m )
: Density of solid phase (kg m )
: Stefan - Boltzmann constant
(5.67 X lO"®)
: Turbulent prandtl number
: Flow parameter
: Variable/parameters in their
equivalent dimensionless forms
Subscript
E : The east neighbouring nodal point
(i.e. in the vertical/ axial plane)
N : The north neighbouring nodal point
(i.e. in the transverse/radial plane)
P : The central nodal point
S : The south neighbouring nodal point
(i.e. in the transverse/radial plane)
E : The west neighbouring nodal point
(i.e. in the vertical/axial plane)
A STUDY ON FLUID FLOW. HEAT TRANSFER. MORPHOLOGY AND
MACROSEGREGATION IN CONTINUOUS CASTING OF STEEL
A Thesis Submitted
In Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
by
SHIV KUMAR CHOUDHARY
to the
Department of Materials and Metallurgical Engineering
Indian Institute of Technology, Kanpur
September, 1993
SYNOPSIS
Continuous casting (CC) , is a process of casting molten
metal continuously. It has over the years gained considerable
importance in both ferrous as well as nonferrous metal
industries. Because of several advantages, continuous casting is
steadily replacing ingot casting throughout the world. Since
1980s, rapid developments in instrumentation and control systems,
availability of superior quality materials, together with
tremendous economic advantages of continuous casting led to its
rapid adoption around the globe. Today, continuous casting is
considered to be the most significant development of the recent
decades in the field of iron and steelmaking • Of the total,
annual World production of about 800 million tons of crude steel,
approximately 55 pet, is now-a-days continuously cast.
XXV
In India, continuous casting of steel made its inroad during
1960s. However, its present share is only about 35 pet. of
approximately 17 million tons of crude steel produced in this
country in 1992. Recognizing the global trend and the merits of
continuous casting, India is trying to make major strides in this
direction. The projected estimates of steel production is about
30 million tons of crude steel by 2000 A.D., and out of this,
more than 90 pet. has been planned to be cast continuously.
Therefore, in future, continuous casting will be the major route
of steel casting in India. To meet such a challenging growth
rate, considerable amount of indigenous research and development
activities are clearly warranted. However, so far, efforts in
this direction in India has not been adequate.
^In the present study, the following important aspects of
continuous casting of steel have been selected for investigation:
(i) mathematical modelling of heat transfer and
solidification during continuous casting of steel,
(ii) morphology and macrosegregation in continuously cast
steel billets,
and (iii) correlations amongst the above to the extent
possible.^
The continuous casting process involves extraction of heat
from the liquid steel. This consists of removal of superheat from
the liquid steel entering the mould, latent heat released during
solidification, and finally the sensible heat of the solidified
metal. ^ Heat is extracted from a solidifying casting by a
combination of several coupled mechanisms such as.
xxvl
(i) convection and turbulent mixing in the liquid pool,
(ii) conduction of heat in the solidified region,
(iii) external heat transfer from the surface of the cast
section by combined mechanism of conduction,
convection, and radiation.
Broadly two different concepts have so far been applied to
model the heat flow phenomena within the liquid pool, viz.
(i) artificial effective thermal conductivity model
and (ii) conjugate fluid flow heat-transfer model.
The former class of model is based on the concept that
convective and turbulent transport of heat in the liquid pool of
a solidifying casting can be represented adequately, if the
central core of the liquid metal is treated like a pseudo-solid
having a relatively large thermal conductivity. While in the
latter approach, the influence of fluid flow on convective and
turbulent transport of heat is described relatively more
precisely via the Navier-Stokes equation in conjunction with an
appropriate thermal energy transport equation.
Of these two approaches, the former has been relatively more
common. Model based on this approach leads to a single conduction
type heat flow equation. In contrast, model based on conjugate
fluid flow and heat transfer phenomena, though has a more
fundamental basis, has been less common owing to the inherent
difficulties in describing the fluid flow and the associated
turbulence in the liquid pool of the solidifying casting. The
latter approach also involves more extensive computational task
as compared to the former.
xxvii
In the present study, mathematical modelling was carried out
by both the above mentioned approaches. Various assumptions were
tested. Also, their predictions were compared with experimental
measurements of shell thickness reported in literature.
Macrosegregation is inhomogeneity in chemical composition
over a macroscopic area in cast products. Macroscopic transport
of segregated liquid and crystals during solidification gives
rise to the macrosegregated regions in cast sections. In
continuously cast products a high degree of positive
macrosegregation in the central region of cast section
constitutes a major defect. This is known as 'centreline
segregation' . Such chemical inhomogeneities due to
macrosegregation are undesirable as they give rise to nonuniform
mechanical properties and cracks in the finished product.
Equiaxed structure during solidification reduces the extent
of centreline segregation by redistributing the segregated liquid
more evenly in between the equiaxed dendrites. Therefore,
morphology of solidification structure has a special significance
in controlling the centreline segregation during continuous
casting.
The thesis consists of five chapters. Chapter 1 briefly
presents :
(i) importance and advantages of continuous casting
of liquid steel and development of the process
(ii) a brief description of continuous casting of steel
(iii) objective of the present study
(iv) plan of work
xxviii
Chapter 2 is concerned with heat flow model based on the
concept of artificial effective thermal conductivity • It
starts with a review of literature on this topic. In the present
study, a steady-state three-dimensional model has been developed.
On the basis of available literature information, boundary
conditions to the governing heat flow equation have been applied,
and the equation was solved via the control volume based finite
difference procedure. The model is sufficiently general and can
be applied to various geometrical shapes of relevance to
continuous casting of steel. Sensitivity of the predicted results
to various numerical approximations including grid
configurations, as well as to other modelling parameters such as
axial conduction, mushy zone modelling procedure, choice of value
of have been extensively studied. It has been shown that
some assumptions and numerical procedures influence the computed
results significantly. Finally, numerical predictions have been
compared with three sets of experimental measurements reported in
literature on shell thickness in industrial casters. In contrast
to some earlier claims, these indicated only poor to moderate
agreement between model prediction and experimental results.
Chapter 3 is concerned with mathematical modelling based on
conjugate fluid flow - heat transfer approach. It starts with a
review of literature on this topic. In the present study, a
steady state, two dimensional mathematical model of continuous
casting of steel has been developed. Governing fluid flow and
heat transfer equations have been derived and a procedure for
their non-dimensional representations outlined. The growth of
xxi V
solidification front and its resultant influences on the fluid
flow and heat transfer have been addressed, and a new calculation
procedure for incorporating the solidification phenomena into the
mathematical model has been proposed.
Control volume based finite difference procedure has been
applied to solve the governing partial differential equations
together with the associated boundary conditions,’ and towards
this a computer program in FORTRAN-77 has been developed.
Adequacy of several key assumptions applied in deriving the
mathematical model has been assessed. Towards this, modelling of
turbulence, liquid steel flow in the mushy zone, thermal
buoyancy, as well as heat flow across the meniscus have been
studied computationally. It has been shown that except for
approximations applied to the modelling of fluid flow in the
mushy zone, predicted flow as well as temperature fields are
relatively insensitive to the accurate modelling of fluid
turbulence, heat flow across the meniscus etc..
Numerically predicted flow patterns in several billet
casters have been assessed qualitatively against equivalent
studies reported in the literature. Similarly, numerical
predictions and reported experimental solidified shell thickness
were compared for three different casting configurations. These
in general demonstrated reasonable to good agreement between
theory and experiment.
Chapter 4 deals with study of macrostructure and
macrosegregation of continuously cast billet samples of low
carbon steel collected from Tata Steel, Jamshedpur. It includes
xxix
an extensive literature review on the topic. Transverse sections
of billets were examined. Macroetching revealed various zones.
The equiaxed zone was mostly asymmetric around the billet centre.
The area of equiaxed zone decreased with increase in tundish
superheat. Samples for analyses of carbon and sulphur were
collected by drilling at the billet centres as well as at the
columnar-equiaxed transition (CET) boundaries, and analyzed by
automatic carbon and sulphur determinator. Degrees of segregation
of carbon and sulphur (r and r respectively) were correlated to
each other both for the centres as well as for the CET boundaries
of the billet sections. Correlation of r with r for the CET
c s
boundaries agreed closely with that predicted by Scheil's or
modified Scheil's ec[uation. Variations of r and r with
o s
fractional solidification, although did not agree with
predictions of the above, were in qualitative agreement with
actual macrosegregation data reported in literature.
Finally, attempts were made to correlate the experimental
macrostructural and macrosegregation data with predictions based
on conjugate fluid flow-heat transfer model. This was achieved
for the CET boundaries. Since temperature of liquid steel could
be measured only in tundish, the loss of temperature from the
tundish to the mould was estimated for arriving at the correct
pouring temperature for computation purposes. Results of this
exercise demonstrated close agreement of predicted locations of
CET boundaries with the measured values.
This is being taken as additional confirmation of the
reliability of the conjugate fluid flow-heat transfer model
developed during the present investigation. Therefore, it is
XXX
proposed that this model may also be employed to estimate size of
iquiaxed zone in continuous casting of steel.
Chapter 5 presents summary and conclusions as well as
suggestions for further work.
CHAPTER 1
INTRODUCTION
Continuous casting (CC ) , is a process of casting molten
metal continuously. It has over the years gained considerable
importance in both ferrous as well as nonferrous metal
industries. In steel plants, continuous casting provides the
necessary link between steelmaking operations and final stages of
rolling, by producing semi finished products such as, billets,
blooms, and slabs. The major advantages of continuous casting
process lie in the fact that it eliminates the conventional ingot
1 2
casting thereby leading to ' :
(i) higher yield (about 10-15 pet. increase),
(ii) increased productivity,
(iii) superior product quality,
(iv) lower capital and operating cost,
(v) reduced energy consumption due to elimination of
soaking pits and primary rolling mill,
(vi) scope of more automation and better process
control ,
(vii) better working environment.
Evidences in literature indicate that the switching over
from conventional to continuous casting operation can account for
an energy saving of the order of 200 MJ per ton of finish
steel^. Because of these distinct merits, continuous casting
steadily replacing ingot casting throughout the world. Ste
plants of the future, as it is readily evident will embo
2
primarily CC for casting molten steel.
1.1 DESCRIPTION OF CONTINUOUS CASTING OF STEEL
Fig. 1.1 illustrates the major components of a typical modern
continuous casting machine. A variety of designs has been
commercialized and new innovations are taking place continually.
Consequently, the salient features which are common to all
designs are summarized below and shown in Fig. 1.1.
Molten steel, tapped in a ladle or similar transfer vessel
from steelmaking furnaces, is subjected to homogenization
treatment by inert gas purging. Subsequently, it is subjected to
secondary steelmaking treatments such as degassing,
desulphurization etc. The extent of such treatment varies from
plant to plant depending on the requirements and the facilities
available. Finally, the ladle containing molten steel is brought
to the CC shop for casting. Liquid metal from the ladle is poured
slowly into a rectangular reservoir, known as tundish, located
immediately above the casting machine. From the tundish molten
steel is poured into one or more open-ended water-cooled copper
mould (typically about 0.5 - 0.9m length). The flow of molten
steel from the tundish to the mould is regulated by stopper rod
or slide gate arrangements fitted to the bottom of the tundish.
To initiate a cast, a starter (i.e. dummy bar) is inserted
through the bottom end of the mould so that it acts as a false
bottom necessary for the casting operation to begin. Molten metal
from the tundish is slowly poured into the mould up to a desired
level, and immediately after that the dummy bar is gradually
withdrawn. The rate of withdrawal of dummy bar (or solidified
metal) must exactly match with the rate of pouring of liquid
3
Automatic
powder feed
u
Remotely
adjustable
mould
Ladle
Ladle shroud
Tundish
Submerged entry nozzle/shroud
Automatic mould level control
Secondory spray
cooling air mist
dynamic control
Withdrawal unit
Strand heat
insulation
Roll support with split rolls
Fig. 1.1. Schematic of a typical slab casting machine.
4
steel for smooth operation. Uninterrupted pouring of liquid steel
from the top and simultaneous withdrawal of cast section at the
bottom of the mould gives rise to a situation in which the melt
can be cast continuously in the form of one solid piece, which is
subsequently cut into the desired lengths.
As the liquid metal enters the copper mould a thin solid
layer, (i.e. the 'skin') is immediately formed due to the
chilling action of water cooled mould wall. The solidified shell,
because of solidification shrinkage, subsequently gets separated
from the mould surface. To prevent sticking of the frozen solid
shell to the mould wall, the latter is oscillated at a controlled
rate during the casting operation. In addition to this, oils or
low melting fluxes (i.e. mould powders) are introduced into the
mould continuously to lubricate the mould wall and to facilitate
easy withdrawal of the casting. Contraction in volume of the
solid shell gives rise to the formation of air gap between the
mould wall and the casting. Simultaneously, static pressure of
liquid in the molten core pushes the shell to bulge outward and
consequently tends to counteract the solidification shrinkage,
and thereby reduce the gas width. The air gap increases with the
progress of solidification along the mould length and constitutes
about 70-80 pet. of the total resistance to heat flow in the
mould/primary cooling zone.
Fig. 1.2 schematically shows the phenomena occurring during
continuous casting process. It may be noted that solidification
is incomplete in the mould region. Therefore, the solidified skin
must be sufficiently thick so as to withstand the ferrostatic
pressure of the melt in the core of the casting. Immediately
below the mould the casting is cooled by spraying water onto the
Steel from Tundish
Fig. 1.2: Schematic of three zones of heat extraction
during continuous casting of steel.
6
cast surface through a series of spray nozzles, to ensure
complete solidification. This region in the CC machine is
typically called the secondary cooling (or spray cooling) zone.
There the casting is mechanically supported by a series of rolls
(i.e. support rolls) . During casting operation, the cast section
is continuously withdrawn by withdrawal or pinch rolls, located
outside the secondary cooling zone. Beyond the secondary cooling
zone, the casting is cooled in the air mainly via radiation. This
zone is therefore called the 'radiation cooling zone'.
1.2 DEVELOPMENT OF CONTINUOUS CASTING OF STEEL
The idea of continuous casting of steel was originally
conceived by Sir Henry Bessemer during 1860s. He attempted to cast
steel sheet continuously, though could not succeed because of
inadequate technology and materials available at that time. By
1930s, continuous casting of nonferrous metals became feasible,
and later proved quite successful. However, the high melting
point, higher specific heat and lower thermal conductivity of
steel were the main obstacles in the development of continuous
casting technology for steel. The technology was first
commercialized in Germany during 1943 and later adopted by a
number of steel plants in Europe. The growth of CC, however,
remained quite limited for sometime and as of 1975, only 5 pet.
of the total World crude steel production was via continuous
casting. Since 1980s, rapid developments in instrumentation and
control systems, availability of superior quality materials,
together with tremendous economic advantages of continuous
casting led to its rapid adoption around the globe.
Today, continuous casting is considered to be the most
7
significant development of the recent decades in the field of
iron and steelmaking . of the total annual World production of
about 800 million tons of crude steel, approximately 55 pet. is
now-a-days continuously cast (Fig. 1.3). In Western Europe it is
above 70 pet., while in Japan, it is well above 90 pct.^"^. It is
expected that in near future, practically 100 pet. of the total
steel produced World wide will be continuously cast with few
exceptions .
Intensive research and development activities in the area of
solidification processing, equipment design, process upgradation,
and automation etc. have played a vital role in bringing the
continuous casting of steel to its present level. Research and
development activities have primarily been directed towards
productivity enhancement, quality improvement and energy saving.
Such endeavor on the continuous casting of steel has led to a
better understanding of the phenomena involved in CC and
established the links between operation and cast quality,
although much remains to be learned.
Towards the success of continuous casting technology,
secondary processing (i.e. ladle metallurgy) of liquid steel has
played an important role. Close control of superheat of liquid
steel is one of the vital requirements for smoother operation and
quality of the CC products. Such requirements can be conveniently
achieved through both homogenization and proper ladle treatment.
In continuous casting plants homogenization treatment of liquid
steel is carried out on a routine basis. Moreover, secondary
steel processing is required for controlling the oxygen level,
inclusion float-out and modification. In addition,
desulphurization, degassing, alloying, etc. during ladle
8
9
treatment have made it possible to cast almost all grades of
steel through CC route successfully.
In India continuous casting of steel made its inroad during
1960s. However, its present share is only about above 35 pet. of
approximately 17 million tons of crude steel produced in this
country in 1992 . Recognizing the global trend and the merits of
continuous casting, India is trying to make major strides in this
direction. The projected estimates of steel production is about
30 million tons of crude steel by 2000 A.D. , and out of this,
more than 90 pet. has been planned to be cast continuously.
Therefore, in future, continuous casting will be the major route
of steel casting in India. To meet such a challenging growth
rate, considerable amount of indigenous research and development
activities are clearly warranted. However, so far, efforts in
this direction in India has not been adequate.
1.3 OBJECTIVE OF THE PRESENT STUDY
In the present study, the following important aspects of
continuous casting of steel have been selected for investigation:
(i) heat transfer and solidification phenomena during
continuous casting of steel,
(ii) morphology and macrosegregation in continuously
cast steel billets,
and (iii) correlation amongst the above to the extent
possible and justified.
10
1.3.1 Heat Transfer and Solidification During Continuous
Casting of Steel
The continuous casting process involves extraction
of heat from the liquid steel. This consists of removal of
superheat from the liquid steel entering the mould, latent heat
released during solidification, and finally the sensible heat of
the solidified metal. Heat is extracted from a solidifying
casting by a combination of several coupled mechanisms such as,
(i) convection and turbulent mixing, induced in the
liquid pool by the momentum of the incoming
pouring stream, as well as natural convection
arising out of thermal gradients in the liquid
region,
(ii) conduction of heat in the solidified region,
(iii) external heat transfer from the surface of the
cast section by combined mechanism of conduction,
convection, and radiation in the mould as well as
the submould region.
Heat transfer plays a crucial role in the smooth and
efficient operation of the continuous casting process. The
productivity and the quality of CC products depend largely on the
rate and
the manner
in which
heat
is
extracted
from
the
solidifying
casting.
Necessary
minimum
thickness
of
the
solidified
shell to
avoid occurrence
of
break-outs
in
the
submould region as well as the depth of the liquid pool relative
to the metallurgical length of the casting machine depend
predominantly on the heat transfer phenomena.
Recognizing the importance of heat transfer phenomena during
continuous casting process, numerous studies have been carried
11
out in the past to improve upon the design and operation of CC
machine and thereby to gain better understanding of the process
fundamentals. Heat transfer phenomena have been extensively
investigated theoretically. Several mathematical models have been
developed to unscore various thermal phenomena encountered in
continuous casting. Different approaches have been adopted and
hence the models differ from one study to another in their
treatment to describe the heat transfer processes within the
liquid pool as well as across the surface of the solidified
casting. Broadly two different concepts have so far been applied
to model the heat flow phenomena within the liquid pool, viz.
8—10
(i) artificial effective thermal conductivity model
and (ii) conjugate fluid flow heat-transfer model^^'^^.
The former class of model is based on the concept that
convective and turbulent transport of heat in the liquid pool of
a solidifying casting can be represented adequately, if the
central core of the liquid metal is treated like a pseudo-solid
having a relatively large thermal conductivity (e.g.
approximately 5 to 10 times the molecular thermal conductivity of
steel) . While in the latter approach ' , the influence of
fluid flow on convective and turbulent transport of heat is
described relatively more precisely via the Navier-Stokes
equation in conjunction with an appropriate thermal energy
transport equation.
Of these two approaches, the former has been relatively more
common. Model based on this approach, as has been described in
the subsequent chapters, leads to a single conduction type heat
flow equation. In contrast, model based on conjugate fluid flow
12
and heat transfer phenomena, though has a more fundamental basis,
has been less common owing to the inherent difficulties in
describing the fluid flow and the associated turbulence in the
liquid pool of the solidifying casting. The latter approach, as
one might anticipate, involves more extensive computational task
as compared to the former.
1 . 3 . 1.1
(i)
(ii)
(iii)
(iv)
(V)
Plan of work for mathematical modelling of heat
transfer
This includes,
derivation of governing heat flow equations for
the artificial effective thermal conductivity
approach and the conjugate fluid flow - heat
transfer approach,
development of flow charts and computer programs
for the numerical solution of governing heat flow
and/or fluid flow equations,
prediction of temperature and velocity profiles in
billet casters,
prediction of thickness of solidified shell from
temperature profile and comparison of the same with
experimental measurements reported in the
literature,
testing of principal assumptions.
and finally,
(vi) assessment of the adequacies of the two modelling
concepts with reference to the mathematical
modelling of heat flow in continuous casting of
steel .
13
1.3.2 Macrosegregation and Morphology in Continuously
Cast Products
Macrosegregation is inhomogeneity in chemical
composition over a macroscopic area in cast products. During
continuous casting, solidifying dendrites reject solute elements
(e.g. C,S,P,Mn etc.) at the solidification front and in the
interdendritic regions. This leads to the gradual enrichment of
residual liquid with progress of solidification. Macroscopic
transport of segregated liquid and crystals during solidification
gives rise to the macrosegregated regions in cast sections. In
continuously cast products a high degree of positive
macrosegregation in the central region of cast section
constitutes major defect. This is known as 'centreline
segregation'. Such chemical inhomogeneities due to
ma'crosegregation are undesirable as they give rise to nonuniform
mechanical properties and cracks in the finished product.
Equiaxed structure during solidification reduces the extent
of centreline segregation by redistributing the segregated liquid
evenly in between the equiaxed dendrites. Therefore, morphology
of solidification structure has a special significance in
controlling the centreline segregation during continuous casting.
Low superheat casting and electromagnetic stirring promote
equiaxed solidification and thereby reduce centreline
segregation. Suction due to solidification shrinkage and bulging
of strand between the support rolls have been identified as the
main cause of flow of segregated liquid leading to centreline
segregation. Adjustment of roll gap taper reduces the bulging
before complete solidification, and thereby decreases the extent
of macrosegregation. Also, soft reduction in the cross section of
14
slab during final stage of solidification has been found to be
quite effective in controlling fluid flow in the mushy zone. In
addition to these, several other techniques have been developed
for controlling the centreline segregation in CC products.
In view of the adverse effect of centreline segregation on
product quality in CC route, numerous studies have been carried
out. As a result of these, it has now become possible to keep the
segregation level below the desirable limit in common grade
steels. However, for high grade steels (e.g. sour gas resistant
steels) control of macrosegregation is relatively difficult.
Therefore, macrosegregation is still a hot topic of research in
the area of continuous casting.
1.3. 2.1 Plan of work on macrosegregation and morphology
This includes,
(i) collection of billet samples and relevant
data from the CC shop of steel plant,
(ii) cutting thin sections from above billet
samples, and polishing of cut cross-section
(i.e. transverse section) for examination,
(iii) physical examination of transverse section in
both unetched and etched condition to
determine various morphological features,
(iv) chemical analysis of drilled samples to
determine macrosegregation levels in
transverse sections at the centreline as well
as at the columnar-equiaxed transition
boundary.
15
(v) Interpretation of results, including attempts
for some correlation with studies on heat
transfer.
1.4 PRESENTATION OF CHAPTERS IN THE THESIS
This investigation has been presented in the subsequent
chapters of the thesis. Heat transfer study based on effective
thermal conductivity model has been described in chapter 2. Study
based on conjugate fluid flow-heat transfer model is described in
chapter 3 . Chapter 4 presents studies on morphology and
macrosegregation. Literature review as well as summary and
conclusions of study are presented in their corresponding
chapters. Attempts have been made to make each chapter self
contained. Chapter 5 presents sununary and conclusions for the
entire work.
CHAPTER 2
MATHEMATICAL MODELLING OF CONTINUOUS CASTING OF STEEL
VIA ARTIFICIAL EFFECTIVE THERMAL CONDUCTIVITY APPROACH
2.1 INTRODUCTION
Heat transfer in the liquid region of a continuously cast
section is relatively more complex than in the solidified region
due to flow induced in the liquid pool via momentum of the
pouring stream, by the buoyancy driven natural convection, and by
the electromagnetic stirring, if present. Therefore, in addition
to conduction, transfer of heat from the liquid to the solidified
region takes place through bulk convection and turbulence. Exact
modelling of heat transfer in the solidifying casting
consequently requires prior knowledge of flow of liquid steel in
the molten pool. This calls for the solution of complex turbulent
fluid flow equations in conjunction with an appropriate equation
of thermal energy transport.
To avoid the inherent complexities associated with the
numerical solution of the turbulent fluid flow equations,
convective as well as turbulent transport of heat have been
either completely ignored^^ or the pool region assumed to be well
mixed^, in the earlier studies on heat transfer in continuous
casting of steel. A somewhat realistic attempt was made
subsequently by Mizikar^, who took into account the enhanced heat
transfer in the liquid due to convection by artificially
increasing the thermal conductivity of the liquid pool. The
17
approach adopted by Mizikar to model the coupled effect of fluid
convection and turbulence on heat transfer in such an ad-hoc
manner has been termed in the literature as the 'artificial
effective thermal conductivity approach'. It has been argued^®
that such an adjustment of thermal conductivity values within the
licjuid pool (e.g. to account for the turbulence and convective
heat transfer in the liquid) is not likely to affect the overall
prediction of the rate of solidification and the temperature
fields in CC, since convection predominantly affects the rate at
which superheat is removed from the liquid steel, and typically,
superheat constitutes only a small fraction of the total heat
content (i.e. latent and sensible heat) of the liquid steel.
The artificial effective thermal conductivity model,
Q
originally proposed by Mizikar , is thus based on the key
assumption that the convective and turbulent transport of heat in
the liquid pool of a solidifying casting can be represented
reasonably well, if the liquid core of the cast section is
considered to be a pseudo-solid having a relatively large thermal
conductivity value (viz., effective thermal conductivity) than
that of the solidified steel. In essence, this implies that the
governing equation describing the flow of heat within CC section
will therefore be a pure conduction type equation. However, it is
to be recognized here that the governing heat flow equation
incorporating the effective thermal conductivity ^^eff^
parameter, the numerical value for which was derived by Mizikar
through force fitting the model predictions against those
measured experimentally on a slab casting machine, lacks a sound
fundamental basis. Indeed, so far there is no explicit evidence
18
in literature to establish that a large thermal conductivity
value arbitrarily assigned to the liquid pool can in principle
accommodate the effect of fluid motion and turbulence on heat
transfer. Despite this limitation, the concept has been
frequently applied by the several subsequent
. . 9 10 17
investigators ' ' , to investigate various thermal phenomena
of relevance to continuous casting.
2.2 LITERATURE REVIEW
7—10
Several mathematical models have been proposed so far to
describe heat transfer and solidification phenomena in continuous
casting of steel. Most of the models are based on the fundamental
equation of heat conduction, and on empirical data to
characterize the complex heat extraction processes across the
surface of the cast strand in different cooling zones. The models
however, often differ from one another in their treatment of the
heat transfer in the liquid pool region.
The initial attempts on the mathematical modelling of
continuous casting were made in the sixties. At that time good
computer facility was almost practically non-existent. Therefore,
7 9 18
investigators ' ' mostly adopted analytical approach to solve
the governing heat flow equation of continuous casting. Towards
this, considerable idealization were made and the most simplified
form of the heat flow equation was considered. One of such
7
studies has been reported by Hills who investigated the heat
7
transfer and solidification in billet casting moulds. Hills
considered a unidirectional transient heat conduction equation as
the governing equation of heat flow and applied integral profile
19
technique to solve the governing equation analytically.
Conduction along the withdrawal direction was ignored and
invariant thermophysical properties of steel were assumed. Molten
steel was assumed to solidify as a pure metal (i.e. at fixed
temperature) , and furthermore heat conduction in the liquid
region was neglected throughout the pool (i.e. well mixed pool
region ) , so that the melt superheat and the latent heat release
following solidification were uniform over the entire pool
region. A constant mould heat transfer coefficient was assumed
and applied as the required boundary condition at the mould metal
interface.
In addition. Hills performed experimental measurements of
solidifying shell thickness over static moulds using the
'pour-out technique' . Predictions were compared with experimental
measurements and reasonable agreements between the two have been
reported. On the basis of mathematical modelling, Hills^ also
analyzed other experimental measurements reported in literature.
In addition to these, a simple heat balance over the mould
cooling water was carried out and a method to deduce the mould
heat transfer coefficients suggested . The mathematical model
proposed by Hills; although was very simplistic, incorporates
several unrealistic assumptions and thus, reliability of
predicted results becomes an issue of concern.
Q
In a SLibsequent study, Mizikar took a relatively more
realistic approach to simulate heat transfer phenomena in
continuous casting and introduced the concept of an enhanced
artificial thermal conductivity to account for the transport of
heat in the liquid pool via convection and turbulence.
20
8
Mizikar was also amongst the first to adopt a numerical
method to solve the characteristic heat flow equation. The
mathematical model proposed by Mizikar is essentially an
unidimensional transient heat conduction equation applicable
strictly to continuously cast steel slabs. An effective thermal
conductivity value equal to 7 times the molecular thermal
conductivity of steel at that temperature, was assigned to take
care of the convection in the liquid pool region. Conduction of
heat along the axial and one of the transverse directions (i.e.
longer side) were neglected. Moreover, latent heat of
solidification was taken into account by adjusting the specific
heat value over the range of solidification temperatures.
Explicit finite difference numerical method was adopted to solve
the governing equation. The Savage-Pritchard correlation ' was
applied to estimate the mould heat flux, and incorporated in the
model as an appropriate boundary condition across the mould wall.
Finally, predicted solidified shell thickness was compared
Q
with the experimentally measured data . A reasonable agreement
between the two has been reported. The appropriate value of the
enhanced thermal conductivity, K^^^(i.e. = 7K) , applied to the
model was deduced by comparing numerical prediction with
experimental measurements. Consequently, the value of in the
liquid pool as suggested by Mizikar is in reality an empirical
parameter. The model developed was further applied to process
design and thus, some optimal set of cooling conditions (viz., an
appropriate spray heat transfer coefficient value) in the
secondary cooling zone was derived by assigning a desired slab
surface temperature all along the spray lengths.
21
2 1
In a separate study, Mizikar investigated experimentally
the influence of spray water flux, spray pressure, spray nozzle
types and their orientations on the cooling characteristics of CC
slab in the secondary cooling zone. Heat transfer coefficients
evaluated as a function of the spray variables were presented as
nomograms. It was suggested that these heat transfer coefficient
data can be applied to the mathematical model as the rec[uired
surface boundary condition in the spray cooling zone.
Consequently, detailed prediction of liquid pool profile, pool
depth, surface temperature in the secondary cooling zone etc. can
be estimated all along the descending cast strand. The data on
heat transfer coefficient were however, limited to a few spray
configurations only.
Gautier et.al^^ also proposed a similar type of model for
predicting the thermal fields in continuously cast square
billets. A transient heat conduction equation in terms of
enthalpy, was considered as the governing heat flow equation.
Although the model was developed in the cylindrical polar
coordinate system, it was applied to predict the temperature
fields in square billets, by reducing the latter to cylinders of
equivalent surface area. In the liquid region convection was
ignored completely, and thus, heat flow in the radial direction
was considered solely by conduction. Furthermore, latent heat of
solidification was estimated from the enthalpy-temperature
relationship. For estimating the heat flux across the mould wall
the mould was divided into two zones viz. the upper contact zone
and the lower gap zone. In the upper contact zone heat flux was
estimated from an empirical heat transfer coefficient data. In
22
the lower gap zone, gap width was estimated first by carrying out
heat balance over the mould, and finally, an expression for the
required mould heat flux was derived, assuming heat flow across
the gap by conduction and radiation. Embodying the resultant
mould heat flux expressions as the boundary condition, the
governing heat flow equation was solved numerically by an
explicit finite difference procedure. Surface temperatures at the
mould exit of a billet caster were measured under a wide range of
casting conditions via a two-color optical pyrometer and compared
against the numerical predictions. The agreement between
measurements and predictions has been reported to be
satisfactory.
Perkins and Irving extended the unidimensional heat flow
8 14
model, reported by the previous investigators ' , further to the
two dimensional situations and introduced other modifications, in
an attempt to make it closer to the real continuous casting
operation. The investigators applied their two dimensional,
unsteady state heat conduction model to analyze the heat flow and
solidification in bloom casters. Effect of mixing and convection
was taken into account by increasing the thermal conductivity in
the liquid region in a manner suggested by Mizikar . Conduction
along the axial direction was ignored and the latent heat release
effect was incorporated by adjusting the specific heat in the
mushy zone. For estimation of surface heat flux, as an
. . 17
appropriate boundary condition in the mould, Perkins and Irving
proposed a three zone heat extraction model for the mould region.
In the upper zone a good contact between the strand and mould
surface was assumed and a higher heat transfer coefficient value
23
applied to the mathematical model. Similarly, in the lowest zone
radiation was assumed as the only mechanism of heat transfer and
correspondingly, a constant lower heat transfer coefficient value
was assigned there. In the intermediate zone, a linear
interpolation of heat transfer coefficient between the values at
the two extremities was considered. Both constant and variable
thermophysical properties were assumed in the computational
procedure, and various numerical techniques were employed to
solve the governing heat flow equation. The predicted shell
thickness and surface temperatures were validated against the
corresponding plant scale measurements. The model was finally
applied to optimize the bloom casting operation.
Q
In their earlier study, Brimacombe et.al have also applied
n
the integral profile technique, originally used by Hills , to
solve the governing heat flow equation (viz., unidimensional
unsteady heat conduction with constant properties) of CC.
Subsequently, based on the artificial effective thermal
conductivity concept, Brimacombe and coworkers ’ ~ carried
out extensive heat transfer studies on continuous casting of
steel. Liquid pool profile and surface temperatures in square
billet casters were predicted numerically and compared with
measured shell profiles. Reasonably good agreement was obtained
only over the upper half of the mould while in the lower half and
the upper spray regions the agreement was relatively less
satisfactory and the measured shell thickness was in general
greater than those calculated. Several factors were analyzed in
order to explain this discrepancy. In this context, adequacy of
several reported surface boundary conditions for the mould
24
region, including the time dependent heat flux correlation
19 20
proposed by Savage and Pritchard ' have been assessed.
However, it was shown that the Savage-Pritchard correlation
describes the heat flux across the mould wall fairly
satisfactorily and therefore, can be universally applied as a
reasonable surface boundary condition in the mathematical model
(in the mould region) . In addition, the investigators^® also
proposed a new correlation between average mould heat flux and
the dwell time of casting in the mould. Furthermore, validity of
the average mould heat flux correlation was verified with the
measured mould heat transfer coefficient values of several
continuous casting machines.
22
Lait and Brimacombe in a subsequent study, solved almost a
similar heat transfer model proposed earlier by Mizikar®, using
the explicit finite difference procedure. Both constant mould
heat transfer coefficient as well as an empirical mould heat flux
correlation were incorporated in the model as boundary condition.
The model was applied to analyze the continuously cast stainless
steel slab and low carbon steel billets. The predicted pool
profiles were compared with the corresponding plant scale
measurements. It has been reported that, while the agreement
between theory and experiment for low carbon steel billets was
quite reasonable, the same for the stainless steel slab was in
general less satisfactory. In addition to this, the
investigators analyzed the validity of various assumptions
viz., mode of latent heat release between liquidus and solidus,
effective thermal conductivity values etc., incorporated in the
model .
25
10 22
In a separate study, Brimacombe modified the model
further and extended this to two dimensional situation in order
to analyze the heat flow in square billet caster more accurately.
In that study, value of effective thermal conductivity in the
liquid was varied between 5 to 10 times the molecular thesnmal
conductivity of steel at that temperature. The remaining features
of the model were however essentially similar to those in the
. 22
previous study . The model was applied to design the mould and
spray cooling configurations (mould length, spray length etc.)
for square billet casting machines. Thus, correlations relating
the working length of the mould, casting speed and shell
thickness were proposed and necessary guidelines for designing
the spray cooling zone to achieve the desired cooling conditions
were also recommended^®.
The effective thermal conductivity approach, originally
8
proposed by Mizikar , has also been applied to study several
p pr ?
solidification related phenomena in CC. Recently, Laki et.al '
have applied the concept to study the microstructural features
such as the dendrite arm spacing and volume fraction of the delta
ferrite in continuously cast stainless steel slab. The
2 6
investigators have also studied solidification in the meniscus
region of a solidifying casting. Satisfactory agreements between
numerical predictions and experimental observations have been
reported.
similarly, Mundim et.al applied Crank-Nicholson finite
difference scheme to analyze heat flow phenomena during slab
casting. The model predictions were used to analyze the
influences of various operating parameters such as melt
26
superheat, casting speed, steel composition, secondary cooling
water flow rates etc. , on the rate of solidification of liquid
steel .
Although several mathematical model studies have been
carried out using the concept of artificial effective thermal
conductivity, only a few of them have reported a reasonable
agreement between the experimental and the computed results
throughout the pool region. By and large, correspondence between
theory and experiments were somewhat acceptable only in the upper
pool region^® (i.e. the mould region). Disagreement in the lower
part (viz . , the submould region) has sometimes been attributed to
1 0 20 **2 4
the error associated with the measurements as well'*’ '
Recently, Lahiri has suggested, on the basis of
mathematical analysis, that the value of effective thermal
conductivity in the liquid pool should be at least 43 times the
molecular thermal conductivity of liquid steel. This finding ,
however, is in much contrast to the equivalent previous claims of
5 to 10 times of the molecular thermal conductivity^®. Similarly,
Mazumdar et.al ' have reported significant discrepancy between
their model predictions and experimental data in literature on
shell thickness. Using a two dimensional pseudo steady state heat
flow model, the investigators analyzed heat flow and
solidification phenomena mathematically in two different square
billet casting operations. Based on the fundamental analysis,
Mazumdar , however, attempted to explain the possible cause of
such discrepancy between prediction and measurements, and
8 10
attributed these to the basic assumption of the modelling '
itself (i.e prescription of a unique value of effective thermal
27
conductivity throughout the liquid region to account for the bulk
motion and turbulent convection on heat transfer) . Mazumdar ,
went on to propose that a single value of the artificial
effective thermal conductivity, although widely accepted, is
physically unrealistic and consequently, not adequate enough to
describe realistically the heat flow in various industrial
continuous casting process.
Considering such divergent views expressed by the previous
investigators as well as the present world wide interest in an
effective alternative approach of modelling, it is naturally
important to assess the adequacy of the effective thermal
conductivity based model as applied to the mathematical modelling
of continuous casting of steel.
Some assessment of the mathematical model studies have been
10 22 29
attempted by earlier investigators ' ' . However, these were
quite limited in their scope (viz., unidimensional model, model
specific to a particular casting configuration etc) . Moreover,
sensitivity of numerical parameters on computed results were not
assessed. In the present study i therefore, the evaluation of
modelling procedure based on effective thermal conductivity
approach has been carried out in a much more comprehensive
fashion as compared to those carried out earlier. The salient
features of the present investigation are noted below.
(i) A steady state three dimensional heat flow model of
continuous casting has been considered, and a computational
procedure developed for solving the same.
(ii) The generalized heat flow model developed in this study
can be applied to the analysis of practically all continuous
28
casting configurations (billets (square and round) , bloom and
slab) .
(iii) The influence of various numerical approximations
and parameters on computed results were assessed rigorously.
2.3 FORMULATION OF THE GOVERNING EQUATION FOR THE PRESENT STUDY
2.3.1 Assumptions in Modelling
Heat flow during continuous casting of billets,
blooms slabs etc. involves several complex phenomena such as:
solidification of molten steel over a range of temperature,
non-planer or wavy solidification front due to non-equilibrium
conditions (e.g. rapid rate of growth, jerky withdrawal of
casting strand etc.) at the solid - liquid interface, segregation
of solute elements (e.g. C,S,P,Mn, etc.) resulting into changes
in morphology (i.e. columnar-equiaxed transition) , shrinkage in
the volume upon solidification and so on. Moreover, cooling below
the solidus temperature is associated with solid-state phase
transformations and volume contraction etc. . Finally, bulging of
strand between the support rolls in the secondary cooling zone
and bending of strand etc. are likely to introduce considerable
amount of complexity to any rigorous mathematical analysis of
heat transfer phenomena during continuous casting of steel.
Consequently, in order to describe heat flow within the
solidifying strand during continuous casting mathematically,
following simplifying assumptions have been incorporated in the
present heat transfer model.
29
(i) Effect of fluid turbulence and convection on heat
transfer has been taken into account by
artificially increasing the thermal conductivity
in the liquid pool region.
(ii) Solidification is essentially under equilibrium
condition.
(iii) Solidification front is flat or planer with
respect to the adjacent liquid.
(iv) Dimension of the cast section remains fixed
throughout the process (i.e. bulging and volume
contraction etc. are ignored) .
(v) Meniscus surface is flat (i.e. no surface
disturbance and melt level fluctuation in the
mould) .
(vi) Invariant density and specific heat of steel.
(vii) Except for the latent heat release, heat effects
associated with other phase transformation
reactions (e.g. 5 -ferrite — > austenite, austenite
— > pearlite etc.) have been neglected.
(viii) Effect of segregation, mould oscillation, bending
of strand etc. have been ignored.
(ix) Due to symmetry of heat flow in square sections
(e.g. billet), only a quadrant of its cross-
section has been considered for the heat flow
flow analysis. For slab, a semi-infinite geometry
has been assumed and heat flow through its narrow
face has been ignored. Similarly, for cylindrical
billet, heat flow has been assumed to be
30
independent of the O-direction (i.e. dl/de = 0)
(see later) .
2.3.2 Governing Heat Flow Equation
Heat flow in the solidifying strand in CC is
essentially multidimensional. Thus, appropriate heat balance
under steady state conditions over a small volume element in the
system (Fig. 2.1), in cartesian coordinate, can be represented in
terms of the following partial differential equation:
raT) a f
aT)
a
^ a
[az J az pef f
azj
;^eff axj
ay
[^eff ay]
(W m”^)
... ( 2 . 1 )
The term on the L.H.S. of Eq. (2.1) represents heat flow in
the axial (i.e. withdrawal) direction (Z) due to the bulk motion
(U^) of the descending strand. The first term on R.H.S. is a
conduction term in axial direction, whereas the second and the
third term represent conduction of heat along the transverse
plane (viz., X and Y directions respectively), p, c, K and S
ef f
are density, specific heat and effective thermal conductivity of
4
steel. S is a source term.
Q T TO 0/1
In the previous studies ' ' ” the effective thermal
conductivity values in the liquid pool was varied
arbitrarily between 5-10 times (mostly 7 times) of the molecular
thermal conductivity of steel. In the present study also, the
value of in the liquid region (i.e. where T a been
assumed, as a first approximation, to be equal to 7 times the
32
molecular thermal conductivity of steel. In the solid region
however, was considered to be equal to the thermal
conductivity (K) of steel at that tempera tiire . Similarly, in the
mushy zone, in general the continuum approach has been considered
and the mixture rule [viz., = f K + (1 - f ) K where f
is the fraction of solid] , applied to estimate the relevant
effective thermal conductivity value.
2.3.3 Modelling of Axial Heat Conduction Term in the
Governing Equation
Because of relatively lower thermal conductivity
of steel than other common metals such as Aluminium etc. and
8 10 22—24
faster casting speed, previous investigators have
ignored the heat conduction along the axial direction and thus,
neglected the 32 ) term in the governing equation
(Eq. (2.1)). Since the exact influence of axial conduction on
overall heat flow have so far not been demonstrated explicitly,
therefore in the present work, an axial conduction term has been
included in Eq. 2.1. Incorporation of this term in the governing
equation, as one might anticipate, would require two boundary
conditions along the Z coordinate axis. Thus, one of the
boundaries e.g., the meniscus from region outside the pouring
stream, can either assumed to be at the prescribed temperature
(i.e. Tq) or regarded as completely insulated (i.e. zero heat
flux across the boundary) . At the outflow boundary, zero axial
temperature gradient is commonly applied boundary condition.
However, for a short domain length the axial temperature gradient
33
may not be equal to zero, and hence, the imposed boundary
condition may not be a realistic one. To take care of this, the
outflow boundary was considered to be located far away from the
inlet boundary so that a zero heat flux condition in the former
is physically valid. Consequently, in the present study, a
sufficiently longer casting strand (at least 5m) was considered
as the appropriate calculation domain and zero axial temperature
gradient was prescribed at the exit as the relevant boundary
condition. These will be discussed in detail in a subsequent
section.
2.3.4 Modelling of Latent Heat Release Effect
In the governing heat flow equation (Eq. (2.1)),
the rate of latent heat release per unit volume (S) during
solidification constitutes a heat source term. During
solidification, latent heat (AH^) is released between the
liquidus and the solidus temperatures (i.e. in the mushy zone).
Since AH^ for liquid-solid transformation of steel is significant
(Table 2.2), therefore, modelling of the latent heat release
effect is critical for the accurate prediction of the rate of
solidification and the overall temperature field in CC. The mushy
zone, where the latent heat is released, however, involves
several complex phenomena such as segregation of solute elements,
dendrite growth, columnar-equiaxed transition, flow through
complex inter dendritic channels etc. , and it is not known to
what extent these factors affect the mode of latent heat release
during the solidification of steel. Consequently, some
idealization have been made to describe mathematically. To
34
this end, in a most simplified approach, the specific heat of
steel has been increased linearly in between the liquidus and
g
solidus temperatures to account for the latent heat release . In
some other studies , a known enthalpy-temperature relationship
has been applied to the heat flow equation to take into account
the effect of latent heat. Alternatively, in the mushy zone the
latent heat release can be computed from the distribution of
solid fraction (f ) assuming equilibrium solidification of
s
pc pc
steel ' . In the present study also, the last approach has been
adopted and thus, the volumetric rate of latent heat release has
been expressed as:
^^s . -3
S = pU^AH^ ^ (Wm ■") ...(2.2)
where AH^ is the latent of fusion of steel
The term (Sf^az) represents change in solid fraction (f^)
with progress of solidification in a volume element (see later)
luring its descend through an incremental distance (AZ) in the
axial direction. The solid fraction can be estimated from the
relevant portion of iron-carbon equilibrium diagram (Fig. 2. 2). At
any temperature (T) between the liquidus and solidus
temperatures, for a given initial carbon (C ) content in steel,
O
solid fraction (f ) can be estimated by applying Lever rule to
s
the iron-carbon diagram as follows:
s
= (C, -
Co)/(C, -
. . . (2.3)
[iicjuid composition (C, ) and corresponding solid carbon contents
Temperature ,
35
Fig. 2.2. Relevant section of the idealised
iron-carbon equilibrium diagram.
36
(C ) in Eg. ,(2.3) have been deduced from the following
expressions, assuming a linear variation of liquidus and solidus
temperatures with pct.C in the phase diagram (Fig. 2. 2).
(T - ...(2.3a)
and Cg = (T - /3g)/ag ...(2.3b)
where a*s and ^'s are the slopes and intercepts of respective
liquidus and solidus lines in Fe-C diagram (Fig. 2. 2).
2.3.5 Boundary Conditions
A schematic representation of the calculation
domain and the relevant boundary conditions have been presented
in Fig. (2.3). These are summarized below mathematically as :
(i) at the meniscus (Z=0)
(a) inside the pouring stream
0 s X s r^ and/or o s y s r^, t = ...(2.4a)
(b) outside the pouring stream
r^< X s a/2 and/or r^< Y s b/2, q^= 0 ...(2.4b)
(ii) at the outflow boundary (Z=L)
0 s X s a/2 and/or O s y s b/2, 5T/3Z= 0 ...(2.5)
(iii) at the axis of symmetry and/or central plane
X=0, 0 s Y s b/2, 0 s Z s L, dT/aX= 0 '
...( 2 . 6 )
Y=0, 0 s X s a/2, 0 s Z s L, dT/dY= 0
(iv) at the cast surface,
X=a/2, 0 S Y S b/2, 0 S Z S L, q = -K
® X=a/2
37
Fig. 2.3. Schematic of the calculation domain in two
dimension and the associated boundary conditions
applied to solve Eq. 2.1.
38
Y=h/2, 0 s X s a/2, 0 a Z s L, ^ BYI
. . . (2.7)
The boundary condition (i) originated from the fact that the
temperature inside the pouring stream was assumed to be the same
as the pouring or casting temperature (T ) of the steel. Outside
O
the pouring stream, melt surface was assumed to be covered with
an insulating slag layer. Therefore, heat flux in this region
across the meniscus can be assumed to be zero. Far away from the
meniscus, the out flow boundary has been considered and the
normal temperature gradient at this boundary was prescribed to be
zero (i.e. b.c.(ii)). Across the central plane, due to
symmetrical heat flow in all directions, zero normal temperature
gradients was assumed (i.e. b.c. (iii)). At the cast surface,
extraction of heat from the surface is complicated by the
formation of an insulating air gap between the cast and the mould
surfaces. Air gap constitutes about 70-80 pet. ' of the total
heat transfer resistances in the mould. The gap characteristics
are complex and often unknown. Heat flux at such a boundary has
been estimated either from the empirical data of mould heat
transfer coefficient or via semi-empirical correlations. In this
. 20
regard, the Savage-Pritchard correlation for the instantaneous
mould heat flux has been reported to give a fairly reasonable
estimate of the mould heat flux and it has been applied by many
previous investigators ' . In the present study also the
Savage-Pritchard correlation has been adopted to deduce the
instantaneous mould heat flux (q_) .
39
Thus, q (Eq. (2.7)) can be quantified in the mould region
s
(OssZ£L,L <L)as;
m m
qs = = [2.67 - 0.33 ] X 10^ ...(2.8)
or, ‘Ija = 1^2.67-0.33 v/V j X 10^ . . . (2.8a)
where t is time in seconds.
In the secondary cooling zone (L^< Z s l^, Ijg^L) , heat extracted
from the surface of the casting is predominantly by impinging
water sprays. Therefore, the surface heat flux (q^) can be
expressed by the following expression :
q = h (T - T )
^s s ' s w^
. . . (2.9)
Similarly, in the radiation cooling zone (L <Zsl) , heat loss from
the casting to the surrounding is purely by radiation and hence,
surface heat flux can be approximated by the following
expression:
% = o-e (0g - 0^) ... (2.10)
The governing equation together with the boundary conditions
summarized above represent the complete mathematical description
of heat flow in CC, which on solution would provide a complete
three dimensional temperature field. Consequently, solid shell
thickness, surface temperature etc., in the cast section in the
different cooling zones i.e. mould, spray and radiation, can be
conveniently estimated.
43
wise profile of the dependent variable (i.e. T) between the nodal
points has been assumed. The numerical integration procedure
involved deriving the volume integral of each terms of Eg. (2.1)
over the control volumes under consideration (e.g. P in Fig. 2. 5),
which leads to the following expression:
/.b ^e
[ f ("oP =(11])'*='
sJw''
/.b
dz =
f f (lz(''ef£ 11]'*=' '*y
S-'w*'
dz
t^
,b ,n ^e
,.b .n ,e
f f 11]]'*="*^ '*" -^ f f [ (lY(‘'eff 1?]'*=' '*y ■*"
J gJ
pb
r^r
+
t'
S’'
S dx dy dz
. . . (2.11)
In Eg. (2.1) it is readily seen that along the X and Y
directions, there are only second order derivatives (i.e. the
elliptic terms) . Numerical integration of these derivatives can
be more conveniently carried out and represented as follows:
b ,n ^e
(lx(*'e« 1^]]
b„n ,e
dx dy dz +
s-'w'
t'' S'* w
(lY(*'e££ 11]'*=' '*y '*=
= [■'eff S £ hft ^ ]"
w s
AX AZ
= L- % + AY AZ +
[- <3n *^3 J
K
eff ,e
K
5x.
<^E -
Tp)
AY AZ +
eff ,w
5x.
w
(Tp -
T„) AY AZ
44
+
K
eff ,n
5y.
n
(Tj^ - Tp) AX AZ
+
^ef f , s
Tg) AX AZ
. . . ( 2 . 12 )
In the axial direction, Z, however, in addition to the usual
second order derivatives, there is a first order derivative
associated with the bulk motion term (i.e. parabolic term) as
well. Consequently, a procedure such as the fully implicit
marching integration, can not in principle be applied to
30
numerically integrate the derivatives along the Z coordinate
It would have been possible if there would have been the first
order derivative only in Eg. (2.1) (i.e. zero axial
8 10
conduction) ' . Thus, the first order and second order
derivatives in the Z-direction were tackled by considering the
former as a convection and the latter one as a diffusion term.
These terms were then numerically integrated using the concept as
applied to a combined convection-diffusion problem proposed by
30
Patankar , as follows:
m ^e
t^
S''W'*
dx dy dz
bulk convection term
W**
diffusion / conduction term
... (2.13)
am "
' (pCn^Tj, - pcuj^) 4X 4V - [ If ]^AX AY
. . . (2.14)
45
In the CC situation, the Peclet number (Pe = pCU^/(K/5)) is
much larger than unity (i.e. a convection dominated case).
. . 30
Therefore, upwind difference scheme (UDS) was employed to
define the convective contribution and the routine central
difference scheme (CDS) for the diffusive (or conductive)
contribution to heat transfer. These concepts transform Eq. (2.14)
in the following form;
pCU^(Tp - T^) AX AY - ^ ^ft^ (Tp - Tp) AX AY
f f t
' S\ ^'^P " V ...(2.15)
Finally, numerical integration of source term (S) yields;
•b ..n ..e
t-' s-'w-'
b
dx dy dz = pU^AH^[fg AX AY
- - *s,t] “
... (2.16)
Substituting, the various terms after integration in
Eq. (2.11), and rearranging, the following discretization equation
can be derived for the governing heat flow equation;
ApTp AgTg + + •^e'^E ^ ^ ^S^S •••(2.17)
Ap in Eq. (2.17) represents the center point (i.e. P) coefficient
of the discretization equation and is defined as;
Ap ^Ap + A^ + Ag + A^ + Ajj + Ag-Sp
. . . (2.18)
46
In Eqs.(2.17) and (2.18) the coefficients, Ag, A^, contain
the contributions of both bulk convection and diffusion from the
neighbouring top and bottom control volumes (i.e. B and T) to the
dependent variable (Tp) at a given central node P. Whereas, the
other coefficients, e.g. A^, A^, etc., contain only diffusion
contribution of the neighbouring control volumes (i.e., E, W,
etc.) to the center point temperature, Tp. The appropriate
expressions for various coefficients can be summarized as:
A,j.
K
SZ
AX AY
K
8X
t— AY AZ
AY AZ
AX AZ
K
eff ,s
6Y.
AX AZ
. . . (2.19)
... ( 2 . 20 )
. . . ( 2 . 21 )
. . . ( 2 . 22 )
. . . (2.23)
. . . (2.24)
Sy in equation (2.17) represents the constant part of a
general linearized source (viz., S = + Sp Tp ) . In the portion
of the domain containing mushy zone, Sy was set equal to the
discretized latent heat source term [i.e. rate of latent heat
evolution in a given control volume] , which is defined as :
fb
r^r
J S''
dx dy dz » Sy = p V“f(^s,b ■ ^s,t) ...(2.25)
Evidently, for the present problem, Sp was considered to be zero.
47
Thus, in a system of n control volumes, n numbers of similar
algebraic equations (viz., Eq. (2.17)) were obtained via the above
mentioned discretization procedure. Also, since the
discretization equations were obtained from the same governing
equation (i.e. the energy balance equation Eq. (2.1)), the former
therefore embodied the same conservation principle as the latter
one. It is interesting to note here that this is an important
feature of the control volume based numerical procedure in
contrast to the routine Taylor series based numerical procedure
(e.g. finite difference technique). The control volume face
conductivities (e.g. ^ t ^ ■" -^etc.) can be computed
taking either arithmetic or harmonic mean (interpolation) of the
relevant conductivity values prevalent at the adjacent nodal
points. As seen from Eqs. (2.19) through (2.24), coefficients of
discretization equations were computed on the basis of the
conductivity values at control volume face as well as the
geometrical features, the latter essentially deduced from the
grid layout applied to the numerical solution scheme.
Prior to the solution of the discretization equations the
boundairy conditions were also transformed into equivalent
numerical form. For implementation of boundary conditions
numerically, only those control volumes located at the domain
• . 30 31
boundaries were considered ' . Discretization equations are
derived via the same above mentioned procedure. In essence,
implementation of boundary conditions were the modification of
either Ap and/or terms of the discretization equations of the
boundary control volumes.
To illustrate this further. Fig. 2. 6 schematically presents
48
the heat flow situation at the boundaries in a 2D calculation
domain (i.e. transverse~X and axial-Z directions only) . The
boundary conditions to the energy balance equations in transverse
direction as has been mentioned earlier were specified via the
surface heat flux expression (e.g. Eqs. (2.7) -(2. 10) ) . Thus,
integration of the X-directional heat conduction term of the
governing partial differential equation for the boundary control
volume (Fig. 2. 6(a)) yielded:
tJ
■b ..n ,e
f f (ixKff 11])’*=' '**' “ [
sJ W''
eff dX
K
eff
dT
dX
J
AY AZ
= [- ‘^w ] ... (2.26)
noting that at the domain boundary :
q_ = known = q ...(2.27)
6 S
(e.g. for the mould region, q is prescribed via the mould heat
flux expression (^, (Eq. (2.8)). Thus, following the procedure
outlined above it can be shown that, integration of the governing
equation around the near surface nodal points (except the
corners) leads to a discretization equation of the type:
‘•b
Temperature
prescribed
(Tt=To)
(c)
W-West boundary
(Axis of symmetry)
Fig. 2.6.
49
T ~ Top boundary
(Meniscus)
E - East boundary
(Cast surface)
r-Vf
Qg prescribed via
2 . 67 - 0 . 337^0
(a)
specified
ds * hs^^s”
(b)
B — Bottom boundory
(Outflow boundary)
cal boundary control volumes in a
calculation domain .
50
Vp ’ ■^b’^b + ^t'^t Vw + Vn *s'^s ^ ( Sa - <*3
. . . (2.28)
in which, Ap = Ag + A,J, + + A^J + Ag
For the secondary cooling zone, during each iteration, surface
temperatures (Tg) of the cast section have been estimated first
via proper extrapolation (higher or lower order methods, detailed
later) of computed internal temperature fields, and subsequently,
surface heat fluxes (q ) were estimated (Fig. 2. 6b) and
substituted in Eq. (2.28). After taking into account the effect of
surface boundary condition via modifying the term as mentioned
above, it is readily seen that the east neighbor 'E' (Figs. 2. 6a
and 2.6b) has no role to play and thus, is isolated from the
calculation scheme.
Similarly.
the prescribed
temperature
(V
inside
the
pouring stream
(Eq.(2.4)) has
been taken
into
account
by
redefining the and Sp terms, in the discretization equation of
the control volumes lying in the immediate vicinity of the
meniscus (Fig. 2. 6c), as follows :
ApTp = + ^e'^E ''' ^n'^N ^s'^s
. . . (2.29)
Similarly, it can be shown that the zero heat flux or zero
temperature gradient at the axis of symmetry (Eq.(2.6)), outflow
boundary (Eq. (2.5)), and at the meniscus (Eqs.(2.4 and 2.4a)) can
be conveniently incorporated considering the coefficients,
A^ « 0, Ag = 0, and A^ = 0 respectively, in the discretization
51
equations of the respective west, bottom, and top boundary
control volumes of the domain (Fig. 2. 6).
After incorporating the boundary conditions, via the above
mentioned procedures, the resultant set of discretization
equations were solved using the well known Tri-Diagonal Matrix
Algorithm (TDMA) adopting a line by line solution procedure. In
this, a particular grid line, say in Z-direction, is chosen and
assuming the dependent variable (T) to be known (viz., guessed)
in the X and Y directions, the problem is essentially reduced to
a pseudo one dimensional situation and subsequently solved by the
TDMA . This was applied to all the grid lines in one direction
and the entire process was repeated for the other two space
directions to obtain a tentative distribution of the
3D- temperature field. This typically constituted one iteration.
The total number of iterations required was decided by the
convergence criteria adopted, which in the present study was
defined according to;
in{Vp- (XWnb * » lo"* ...(2.30)
in which,
Z^nb'^nb ^ ^"^T *w'^W
... (2.31)
The triple sum in Eg. (2.30) represents the summation over the
entire volume (e.g. the calculation domain) . A typical under
relaxation factor of 0.2 on the dependent variable has been
employed in all the computations to achieve/enhance the
convergence. CENTRAL
52
2.4.2 The Computer Program
For the numerical solution of the present problem,
a general computer program in FORTRAN 77 and in double precision
has been developed. The program is so written that three
dimensional computations (3-D) as well as those in 2-D and 1-D
can also be performed by manipulating certain key parameters.
Furthermore, an interesting feature of the present computer
program has been that, the same program can also be used for
computations in cartesian as well as in the cylindrical polar
coordinate systems. The transformation from one coordinate system
to another or from one geometry to another can be illustrated by
considering a general form of the heat conduction equation
presented in Section 2.3.2:
P=“o(S) " IzKff al] ^ i) a?) * s
... (2.32
in which, has been defined as an index of coordinate system
and
(i)
^2 as index of coordinate
dimension.
if.
X^“l and ^ 2 — Ip
(ii) if, and ^ 2=0
(iii) if, Xj^=r=X and ^ 2 “®'
Eq. (2.32) becomes a 3-D heat
conduction equation in cartesian
coordinate, applicable to heat flow
during billet casting
gives 2-D heat conduction equation
in cartesian coordinate, applicable
for slab caster
represents heat flow in 2-D
cylindrical polar coordinate,
applicable to round/ ax i symmetric
billet caster.
53
As mentioned already, all the above three types of typical
casting geometries have been simulated computationally in the
present study.
The computer program consists of several subroutines or
module for each specific operation. Flow chart of the program is
shown in Fig. 2. 7. A typical computation is initiated with
specifying the coordinate system and dimension (i.e. 3-D or 2-D
and cartesian or cylindrical-polar) . Relevant data and grids in
various directions are specified. Geometric quantities, viz.
distance between nodes and control volume faces, area of control
volume faces and their volumes etc., are computed and conditions
at each nodes are initialized. Subsequently, the following
sec[uence of operations are carried out during each iteration till
a converged solution obtained.
(i) The thermophysical properties are updated in subroutine
PROPS ,
(ii) coefficient of discretization equations are calculated
in subroutine CALCT,
(iii) boundary conditions are incorporated in subroutine
BOUND,
(iv) coefficients of discretization equations, are
reassembled in subroutine CALCT,
(v) the system of discretization equations are solved via
TDMA in subroutine LISOLV,
(vi) sol id- fraction at various locations computed in
subroutine FEC,
(vii) surface temperature is estimated from the predicted
internal temperature field.
54
Fig 2.7. Flow chart of computer program for the model based
on effective thermal conductivity concept.
55
(viii) the above mentioned steps (i) through (iv) are
repeated till a converged solution is obtained,
(ix) from the converged solution (i.e. the temperature
field) final cast surface temperature and shell
thickness, for various axial positions, are estimated,
(x) relevant out put data are printed in subroutine PRINT.
All computations were carried out on the HP-9000 super mini
computer available at I.I.T. Kanpur. As summarized in the
subsequent sections, relevant data of actual CC operation were
taken from literature, for carrying out numerical computations
and their subsequent validation with experimental measurements.
2.5 RESULTS AND DISCUSSIONS
2.5.1 Sensitivity of Computation to the Choice of Grid
Distribution
A variety of grid systems were employed in order
to arrive at the practical grid independent solutions. Figures
2.8 and 2.9 respectively show the variations of shell thickness
and midface surface temperature with distance below the meniscus
for the various grid configurations tested. These further show
that 25 X 40 and 25 x 80 produced almost identical estimates of
56
Fig. 2.8. Variation of shell thickness with distance below
meniscus for different grid configurations
(data set 4 , table 2.1 ).
Surface temperature.
57
Fig. 2.9. Variation of surface temperature with distance
below meniscus for different grid configurations
(data set 4, table 2.1).
58
both shell thickness as well as surface temperature. However,
minor differences existed between 25 x 20 and 25 x 80 grids or
for 16 X 40 and 25 x 40 grids. These appear to indicate that for
25 X 40 grids the solution became nearly grid independent.
However, the solution, as reflected from these figures seem to be
relatively more sensitive to the number of grid points along the
transverse directions, which is evident from the differences
indicated between predictions derived via 16 x 80 and 25 x 80
grid systems. Other grid configurations (viz., 12 x 80, 18 x 80,
25 X 100 etc.) were also tried. However, 25 x 40 grid
configuration, equivalent to a grid spacings of 3 mm in the
transverse and 50 mm in the axial direction, was found to be
satisfactory for arriving at grid independent solutions from a
practical stand point. Consequently, similar grid spacings (e.g.
transverse ss 3mm and axial a 50 mm) as those corresponding to 25
X 40 grid systems were employed in all subsequent calculations
reported.
2.5.2 Influence of Various Numerical Approximations on
the Computed Results
2.5.2. 1 Arithmetic mean vs. harmonic mean approximation
for estimating the control volume face thermal
conductivity
Before computations with actual CC data are
carried out, and the results compared against the experimental
measurements, influence of various numerical approximations on
the predicted results were rigorously assessed. In the present
59
study, as has been mentioned already, liquid region was assumed
to have a higher thermal conductivity (seven times the molecular
themal conductivity, K) than that of the solidified region.
Similarly, the temperature dependent thermal conductivity of
steel led to a highly nonuniform distribution of thermal
conductivity in the calculation domain. While thermal
conductivity values were known only at the grid points,
calculation of coefficients of discretization equation
(Eqs. (2.18)-(2.23) ) required thermal conductivity values to be
known at the mid positions between the nodal points (i.e. at the
control volume faces) . Consequently, it was necessary that proper
interpolation techniques were applied to estimate thermal
conductivity at the control volume face from those of the
adjacent nodal points.
In the present study, control volume face conductivities
were computed via (i) arithmetic mean and (ii) harmonic mean
approximation procedures . For a given control volume face (say
'e' in Fig. 2. 5), midway between the nodes (i.e. P and E) ,
arithmetic mean approximation gives the following value of
control volume face conductivity:
eff ,e
1 (■'ef£,P J'eff.E)
(2.33)
In contrast, harmonic mean approximation for the same provides:
K
eff
^ ^eff,P ^ ^eff,E
[^eff,P *^eff,E]
. . . (2.34)
Shell thickness, mm
rig.2.10: Effect of arithmatic mean and harmonic mean approximation
techniques (e.g. for control volume face conductivity) on
predicted shell thickness in o square billet .
(condrtions of computob'ons ore summonaaed in Table 2.1)
Distance below meniscus, m
Fig. 2. 11: Effect of orithmotic meon and harmonic meon approximation
(e.g., for control volume foce conductivity) on predicted
midface temperature in a squore billet
(data set 2, Table 2.1)
62
Figs. 2.10 and 2.11 respectively present the variation of
computed shell thickness and surface temperatures derived via
arithmetic and harmonic mean approximation techniques. Despite
wide variation of thermal conductivity in the calculation domain,
these show practically negligible differences and indicate that
the two procedures provide practically identical estimates.
Therefore, from the view point of relative simplicity the
arithmetic mean interpolation procedure has been adopted for all
subsequent computations.
2. 5. 2. 2 Lower order vs. higher order interpolations for
estimating surface temperatures
As shown in Fig. 2. 6, at the cast surface, there is
no grid point and hence cast surface temperature has to be
calculated from the predicted internal temperature fields via
some suitable interpolation techniques. Similarly, in the
secondary cooling zone, the boundary condition at the cast
surface has been prescribed via the heat transfer coefficient
(h ) and the spray water temperature (T ) . In order to calculate
S w
the surface heat flux (q ) via Eq. (2.9), surface temperatures, at
various positions in the spray cooling zone, have to be estimated
first from the computed internal temperature field during each
iteration. Therefore, accurate estimation of the surface
temperature is critical for the reliability of the predicted
results. The surface temperatures can be estimated from the
internal temperature field using either the lower order or the
32
higher order boundary treatments
63
(a)
Fig. 2.12. Boundary control volumes considered
for (a) lower order and (b) higher order
interpolation methods for estimating cast
surface temperature.
64
In the lower order interpolation procedure, for example,
surface temperature or surface heat flux is estimated on the
basis of a single nodal point temperature adjacent to the domain
boundary (see Fig.2.12(a) ) . Thus, on the basis of such
considerations ,
■Js “ ^ K-1 -
’'s = Vi--r'Js ...(2.35
s
On the other hand, the higher order interpolation takes into
account two successive internal nodes adjacent to the strand
surface boundary (Fig. 2. 12 (b) ) for estimation of the relevant
32
surface temperature or heat flux as follows :
^s
in which.
‘3n-
n-1
K.
^ (^2 - Vl)
. . . (2.36)
. . . (2.37)
As is well known, the higher order interpolation technique is
likely to provide relatively more accurate values of the surface
temperature , since this takes into account the influences of
other neighbouring nodes and consequently, makes the energy
balance physically more meaningful.
Fig. 2. 13 presents the midface temperature variations along
the axial direction, as estimated via the two interpolation
procedures. There, the higher order interpolation method is seen
to predict somewhat higher surface temperature (about 4 pet.)
Distance below meniscus, m
Fig.2.13. Effect of higher order and lower order interpolation
techniques on predicted midface temperature in a square
66
throughout the strand as compared to its lower order counterpart.
Despite such marginal differences, the higher order interpolation
technique has been considered in the present work since it is
physically more realistic than the lower order method.
2. 5. 2. 3 Influence of different numerical integration
procedure for the mould heat flux expression
As pointed out earlier, the instantaneous heat
flux expression represented via Eq. (2.8) has been applied as a
prescribed surface boundary condition in the mould region. In
order to estimate the rate of heat extraction, the heat flux
expression has to be integrated numerically over the control
volume faces at the cast surface boundary (Fig. 2. 14). For a given
control voltime P, as shown in Fig. 2. 14, the heat flux expression
can be integrated numerically via the following procedures:
fb
ff
J gJ
J
dx dy dz = AX AY
%
dz
(2.38)
in which, AX AY represents the control volume face area on which
the heat flux q^ is assumed to prevail.
Furthermore ,
5 pb
<J^ dz = ^2.67 - 0 . 33 /z/V^ ]
t'*
dZ X 10
6
. . . (2.39)
67
Fig. 2.14. A 2D representation of a typical
boundary control volume in the
central vertical plane of a square
billet.
68
The net heat flux can then be estimated on the basis of;
(i) solely the location of central nodal point P (referred
to as route-1), i.e., in terms of the instantaneous distance
(Zp) . Equation (2.40) then becomes:
pb
dz = ja.e? - 0.33 J Zp/ j AZ X 10®
t**
. . . (2.40)
or, (ii) locations of either top or the bottom faces of the
control volume P.
Estimation of the heat flux based on the top face of control
volume gives (i.e. route-2) .
pb
q^jj dz = f2.67 - 0.33^Z^/ U^j AZ X 10®
. . . (2.41)
In terms of a pseudo time coordinate, the above mentioned
estimates of heat flux values can be seen to be explicitly
defined in terms of the previous time step value. Similarly,
those estimated on the basis of bottom (or the leading) face
alone means that the flux is estimated solely on the basis of
the current time step value (referred to as route-3) . Thus, the
corresponding integration procedure of the mould heat flux
expression yields the following expression ;
-b
q^j^ dz = f2.67 - 0.33y^Z^/ir | AZ X 10®
. . . (2.42)
70
In view of routes 2 and 3, route-1 can be visualized to be a
combination of route 2 and 3, and therefore can be considered to
be analogous to a semi-implicit or semi explicit estimation of
the integrated heat flux.
In the present study, instantaneous mould heat flux was
evaluated via all the three above mentioned numerical integration
procedures. Predicted results thus obtained were presented in
terms of variation of shell thickness with distance below
meniscus in Fig. 2. 15, Very little differences, have resulted from
these three considerations as is evident from Fig. 2. 15. From a
theoretical point of view, average distance between the two
control volume faces (i.e. route-1) is more realistic. Therefore,
on the basis of Fig. 2. 15, route-1 was employed in all subsequent
computations for integrating the instantaneous mould heat flux
expression.
2.5.3 Influence of Axial Conduction on the Computed
Results
In the previous studies® ' , conduction of
heat in the axial direction has been assumed to be negligible. It
has been considered in general that relatively higher withdrawal
rate of casting and lower thermal conductivity of steel makes the
term in Eq. (2.1) dominant in comparison to the
corresponding conduction term along the same (e.g. axial)
direction. However, in the present study the axial conduction
term has been incorporated in the governing equation to test
directly the validity of such an assumption. Thus, to assess the
sensitivity of the axial heat conduction term in Eq.(2.1),
71
Fig. 2.16. Influence of axial conduction term in the
governing heat flow equation on predicted
shell thickness
(data set 4, table 2.1).
72
Fig. 2.17. Influence of axial conduction term in the
governing heat flow equation on predicted
temperature
(data set 4, table 2.1).
73
calculations were carried out by ignoring axial conduction
altogether as well as considering axial conduction in the
numerical calculation procedure. Results thus obtained are
svimmarized in Figs. 2. 16 and 2.17 which show only marginal
influences of the axial conduction term on the resultant
predicted shell thickness and surface temperature. On the basis
of these, it can therefore be concluded that axial conduction of
heat has no significant role to play so far as transport of heat
in CC is concerned.
2.5.4 Influence of Modelling Procedures Applied to
Approximate Heat Conduction in the Mushy Zone
Q IQ 22
In some of the earlier studies ' ’ , from the
view point of heat flow, mushy zone has been considered as a
solid region and thus, the effective thermal conductivity =
7K) value was assigned only to the completely liquid region (i.e.
where T > . However, the characteristics of mushy zone is
somewhat different from either the complete liquid or solid
regions. Lait et.al have studied the influence of incorporating
the mushy zone in the liquid region (i.e. = 7K assigned in
the regions T a *^301^ well. About 6 pet. overall increase in
the shell thickness for plain low carbon steel billet has been
reported from such consideration. Some other investigators
assumed the thermal conductivity of mushy zone to vary with
2 c: o £
fraction of liquid at various positions in this zone ' . In the
present study both the previously mentioned approaches were
considered to assess directly the influences of specific
modelling procedures applied to the mushy zone on overall
Shell thickness, mm
Fig.2.18: Influence of mushy zone treatment on predicted
shell thickness in a square billet caster
(data set 2, Table 2.1)
76
predicted heat transfer rates. The mushy zone treatment, in
principle therefore, refers to the estimation of thermal
conductivity in the two phase region via the following
expression, based on the popular mixture model;
^ush = "'s + - *s) *'eff ...(2.43)
In another set of computation = 7K (i.e. without treatment)
has been assigned throughout the regions above the solidus
temperature (i.e. T > steel. This corresponds to a
procedure where mushy zone has been essentially treated as a
liquid.
The results derived on the basis of two such considerations
have been presented in Figs. 2. 18 and 2.19. From these, it is at
once evident that the estimates of shell thickness while are
essentially similar, some differences (about 4-5 pet.) between
the respective estimates of mid-face temperature exist
(Fig. 2. 19). Theoretically, treating the mushy zone as a mixture
of solid and liquid appears to be a relatively better
approximate, and hence this concept was applied to all subsequent
computations .
2.5.5 Influence of Values of Mould Heat Flux on the
Computed Results
2. 5. 5.1 Instantaneous vs. average mould heat flux
expressions as the surface boundary condition in
the mould region
In the present study, as mentioned earlier, the
Savage-Pritchard correlation (Eq. 2.8) for the instantaneous
77
mould heat flux has been applied as the surface boundary
condition in the mould region. Lait et.al^^ have also proposed
another correlation based on the Savage-Pritchard correlation for
estimating the average mould heat flux. Validity of the average
mould heat flux correlation has been demonstrated by the same
investigators through comparison with industrial experimental
data on a wide range of continuous casting machines. According to
Lait and co workers , the average mould heat flux (<^) can be
correlated with the average mould dwell time (tj^^) of the casting
via the following expression:
= [2.69 - 0.223 X 10® ...(2.44)
in which, the dwell time t^ is defined according to:
^m U
m
. . . (2.45)
The sensitivity of the computed results to the choice of the
heat flux expressions viz., Eq. (2.8) and (2.45) as surface
boundary condition in the mould region has been investigated and
the results thus obtained are shown in Fig. 2. 20, where estimates
of two sets of shell thickness have been directly compared.
Evidently, no significant difference was found between the two
sets of estimates. Since the instantaneous mould heat flux
expression is physically more meaningful, consequently this can
be considered to be more appropriate in the numerical solution
scheme .
Shell thickness,
80
2. 5. 5. 2 Confidence limit of mould heat flux expression and
its likely influence on the accuracy of computed
results
Based on several measurements on industrial billet
and slab casters, the mould heat flux correlations have been
shown to be within a scatter of ±50 pet. ' . Calculations have
been consequently carried out by arbitrarily changing the average
mould heat flux value up to about ± 40 pet., and results thus
obtained are presented in Fig. 2. 21. It is evident from the figure
that the actual limit of confidence of the mould heat flux has an
important bearing on the overall predicted results and thus, is
expected to affect the overall accuracy of numerical computation
considerably.
2.5.6 Sensitivity of Computation to the Choice of
Effective Thermal Conductivity Values
In the artificial effective thermal conductivity
Q
model, as proposed originally by Mizikar , the effective thermal
conductivity value applied (i.e. to the liquid region has
been an empirically fitted parameter. Furthermore, there is at
present no evidence of any fundamental nature for the selection
of an appropriate value for varying casting configurations.
Mizikar deduced the value empirically by matching a set of
theoretical predictions with the corresponding measurements on an
industrial slab caster, and found that = 7K in the liquid
pool of a typical slab caster provides reasonably good agreement
between theory and experiments. Brimacombe and coworkers^®
applied value equal to 5-10 K in their studies. However, the
81
value of = 7K in the liquid has been more frequently
employed by the subsequent investigators, to account for the
turbulent and convective transport of heat in the liquid pool
region of a solidifying casting. Contrary to all these, in a
recent study, Lahiri on the basis of purely theoretical
considerations has suggested that the value of should be at
least 43K. In view of a wide range of values suggested in
the literature, an attempt has been made to assess the
sensitivity of computation to the choice of an appropriate value
of the in the liquid pool, so that industrial conditions can
be effectively simulated. In this connection, values of
selected were IK, 7K, 12K and above 30K for • numerical
computation .
A typical slab caster was selected for the numerical
calculation with the above mentioned values of In slab
caster, liquid pool typically is relatively more wide and the
pool depth consequently is shallower than billet or bloom
casters. Therefore, the effect of turbulence and convection is
expected to be more pronounced in slab than those in other
configurations. Fig. 2. 22 presents estimates of shell thickness
for various values, which illustrates that the shell
thickness did not vary much if the is varied between IK and
7K. However, there is significant influence on shell thickness
when value was arbitrarily increased to 12K. It has also
been observed that a value above 3 OK did not give any
solidification at all in the mould region. Equivalent results for
a square billet caster are also presented in Fig. 2. 23. This
however, showed that the differences between predictions via
83
Fig. 2.23. Present estimates of solid shell thickness in a
billet caster for different effective thermal con-
ductivity values and their comparison with
experimental measurements^^
(data set 1, table 2.1).
84
= IK and = 7K were somewhat more pronounced than those
observed in slab caster (viz.. Fig. 2. 22).
The trend in results reflected by the choice of different
values, as presented in Figs. 2. 22 and 2.23 can however, be
rationalized as follows. The overall heat flux at any location in
the transverse direction, say X, can be expressed by the
following relationship.
Qv = - K
dT
eff dX
. . . (2.46)
For Fig 2.22 or 2.23, heat flux at the boundary is the same
for all the curves for the mould region. Therefore the internal
temperature distribution would depend on the magnitude of assumed
value. For lower values (e.g. IK) relatively large
temperature gradients and for higher (e^g* 12K) values more
uniform and hence less temperature gradient are to be expected
within the lic[uid metal pool. Therefore, if the is
arbitrarily made much larger, the transverse or radial variation
of temperature from the center line (nearly equal to the pouring
temperature) can be expected to be only marginal. Thus, for
= 12K, little difference in predicted temperatures between
meniscus and any other locations in the mould region has been
observed. Consequently, such redistribution of temperature
produces very little shell in the mould. In contrast, at =
IK, from a similar stand point, predicted temperature difference
between meniscus and elsewhere was much larger, and thus,
relatively more solidification (e.g* more shell thickness) is
observed computationally.
85
is an adjustable parameter and there is little
information based on which this can be assigned appropriate value
(only scope is trial and error) . Consequently, the most popular
and widely applied value (viz., = 7K) has been adopted in
the present study for computations. It may be stated that this
choice was quite reasonable as well from the point of view of
Fig. 4.23, and the preceding discussions of the same.
2.5.7 Comparison of Results with Experimental Data
from Literature
In order to validate the mathematical model and
thus to test the adequacy of the model developed, comparisons
have been made between predicted results and corresponding
experimental measurements reported in literature. Figs. 2. 2 4 and
2.25 present the computed solid shell profiles for two different
square billet casters. Relevant data employed to these
computations have been obtained from the reported studies by Lait
22
et al . These authors carried out extensive measurement of pool
profiles by radioactive tracer technique on industrial casters.
Table 2.1 presents the data employed to deduce the results shown
in Figs. 2. 24 and 2.25. However, some of the useful information
such as melt superheat and spray heat transfer coefficient, are
not available in Ref. 23. Therefore a typical superheat value of
25°C was employed in computation of results presented in
Figs. 2 . 24-2 . 25. Because of uncertainty in the values of spray
heat transfer coefficient, comparisons have been restricted to
the mould region only, although the secondary cooling zone was
also included (with an estimated heat transfer coefficient data)
Distance below meniscus, m
Fig. 2.25: Comparison between the predicted and experimental
shell thickness for a typical square billet caster
(conditions as in ref. 22 for 0.133 m sq. billet).
88
in the numerical solution scheme. It is important to note here
that in view of negligible axial conduction (as has been shown
already ) , inclusion of secondary cooling zone in the computation
is not likely to affect the predicted results in the mould
region. Thus, Figs. 2. 24 and 2.25 show that the overall agreement
between the experimental and predicted data is somewhat poor and
less satisfactory than those suggested earlier '. The
agreement between theory and experiment was however reasonable in
the upper mould region alone.
In Fig. 2. 26 another set of experimental data for a typical
round billet caster reported by Ushijima , have been presented
along with the present theoretical estimates. Fairly reasonable
agreement for the upper pool region (i.e. mould) is demonstrated
in the plot. Whereas, in the submould region, significant
discrepancy between the experimental and predicted data is
evident. As mentioned already, discrepancy between prediction and
observation can be attributed to the experimental technique (in
this case, visual observations) as well as to the mathematical
modelling concepts applied. Towards this, it is important to
mention here that Asai and Szekely^^ have demonstrated good
agreement with the observation reported by Ushijima via their
fluid flow based heat transfer model. Hence it cannot be stated
that the discrepancy illustrated in Fig. 2. 26 is due to heat
transfer coefficient value employed in the present investigation.
As a final point, it is to be mentioned here that none of
the earlier studies attempted to examine the issues of sources of
uncertainty in the numerical calculation procedure. However, as
the present study has indicated that the numerical approximations
Shell thickness,
Fig. 2.26: Comparison between predicted shell thickness for
various grid configurations and corresponding
experimentoL measurements of a typical round
billet caster“(conditions as in ref.l1).
90
applied have considerable bearing on the accuracy of the
predicted results. Similarly, in no studies reported so far, the
issue of grid independent solution has been addressed.
To illustrate this point further, in Fig. 2. 26 predictions
based on 12x40 and 16x80 grid systems have also been included in
addition to those derived via 25x40 grid system. With reference
to Fig. 2. 26, it is important to note that prediction of shell
thickness are relatively more sensitive to grid distribution in
the transverse/ radial direction. This is to be expected since
variation of thermal properties are relatively more steep in the
transverse direction than are in the longitudinal
direction/withdrawal direction. Considering the pouring
temperature as a reference, immediately below the meniscus the
thermal field derived via a fine grid system is expected to be
relatively more close to the reference temperature than those
deduced via a sparse grid system. Since a higher temperature
field results in a low solidified shell thickness and vice versa,
consequently, it is seen in Fig. 2. 26 that with the increase in
grid configuration in any direction, the shell thickness
decreases. Similarly, predicted surface temperature derived from
internal temperature field is also expected to be somewhat
greater for finer grid system, leading to different heat flux
(=hAT) values at the cast surface for the two grid systems.
Consequently, the influence of grid distribution is seen to be
relatively more pronounced in the secondary cooling zone than in
the mould region.
Figure 2.26 appears to indicate that as the grid
configuration applied became more and more sparse, the prediction
91
tends to come closer to the observation, and this apparently
suggests that reasonable to excellent agreement exists between
theory and experiments. Nevertheless, as has been discussed-
already, the numerical results, unless are independent of nodal
configuration, have no validity and hence comparison of such
results with experiments have in reality, no meaning.
Hence, in Fig. 2. 26 comparison of experimental measurement is
meaningful only with curve 3 (24x40 grid) , for which the solution
was found to be grid independent. This shows that agreement
between predictions and experimental data are poor. The authors
have already discussed it. It may be due to uncertainties in
experimental data or inadequacy of mathematical models based on
effective thermal conductivity concept. It is possible that both
of these are contributing partly to discrepancy. It is not
possible to make any more definite statement without further
studies .
2.6 SUMMARY AND CONCLUSIONS
Based on the concept of artificial effective thermal
conductivity approach a steady state 3D heat flow model of CC has
been developed. Control volume based finite difference procedure
has been employed for the numerical solution of the governing
heat flow ec[uation. A general computer program, which
incorporates Tri Diagonal Matrix Algorithm (TDMA) for the
solution of discretization equation, has been developed in
FORTRAN 77. The program is so written that computations in
cartesian as well as in cylindrical polar coordinate systems can
be performed in both 2-D and 3-D. Before carrying out any
92
comparison between theory and experiments, sensitivity of
numerical solution to grid configuration and various numerical
approximations in the calculation procedure, have been analyzed.
Validity of various assumptions in the modelling has also been
tested and finally, the computed results thus obtained have been
compared with experimental measurements reported in literature.
The present study has revealed that the axial conduction
term has a minor role to play so far as the modelling of overall
heat flow in CC is concerned. Numerical solution has been found
to be relatively more sensitive to the choice of grid
configurations in the transverse direction. Similarly, procedures
applied to model heat flow in mushy zone as well as the surface
boundary condition in the mould were found to affect the
predictions somewhat. In order to select a proper value and
test its sensitivity to computed results, values of were
varied over a wide range. Finally it was decided to take *=7K
for further computations.
Model predictions have been assessed against three sets of
experimental data for round and square billet casters. However,
in most of the cases the overall agreement between predictions
and experimental measurements of shell thickness were not found
to be satisfactory. Validity of other assumptions such as,
equilibrium solidification of steel, constant thermophysical
properties, have been already established by the previous
investigations .
Considering all the points mentioned above, it appears that
the concept of artificial effective thermal conductivity as
applied to the liquid pool to account for the effect of fluid
93
convection and turbulence on heat transfer, is not adequate
enough to describe various thermal phenomena in CC realistically.
To test this hypothesis further, in the next chapter a relatively
more fundamental model based on the concept of conjugate fluid
flow and heat transfer has been presented.
94
Table 2.1; Data of CC used in the present computation
Parameter
Data
Set-1
[ref .22]
Data
Set-2
[ref .22]
Data
Set-3
[ref .27]
Data
Set-4
[ref .33]
Cast geometry
square
square
slab
round
Cast size (m x m)
0.14
0.133
0.25
0.115
Mould length (m)
0.51
0.685
0.6
0.5
-1
Casting speed (ms )
0.0254
0.044
0.0125
0.0317
Steel carbon (pet.)
0.1
0.1
0.14
0.1
Melt superheat* (°C)
25
25
25
25
Solidus temp. (°C)
1496
1496
1496
1496
Liquidus temp. (°C)
1529
1529
1526
1529
Spray heat transfer
-
-
—
1079.45**
coefficient (W m ^ °C
1"" — ",
* - estimated
*» - source; ref. [11]
Table 2.2; Thermophysical properties of steel used in
computation
Density
1 -3
kg m
7400
Latent heat of
solidification
J kg ^
271954
Specific heat
J kg ^ °C ^
682
Thermal conductivity
W m“^ °c“^
34.6
&
15.89+0. OUT
* ~ source; Physical Constants of Some Commercial Steels at
Elevated Temperatures, Ed. The British Iron and
Steel Research Association, Butterworths Scientific
Pub., London, 1953.
** - source; Ref. [8]
CHAPTER 3
MATHEMATICAL MODELLING OF HEAT TRANSFER IN CONTINUOUS CASTING
OF STEEL VIA CONJUGATE FLUID FLOW AND HEAT TRANSFER APPROACH
3.1 INTRODUCTION
Continuous casting of steel involves coupled fluid flow and
heat transfer phenomena. Thermal energy from/within the molten
core of a solidifying casting is transported by turbulent mixing,
convection, as well as by the molecular thermal diffusion (viz.,
conduction) . The rate of this turbulent convective transport of
heat is governed principally by the kinetic energy of the pouring
stream as well as by the buoyancy induced natural convection.
Therefore, from a fundamental point of view, detailed
mathematical modelling of the continuous casting process
necessitates solution of fluid flow eguations concurrently with
an appropriate heat transfer equation. Thus, the mathematical
model reported in the present chapter has been referred to as the
conjugate fluid flow and heat transfer model of continuous
casting process. The model, in its detailed form as will be
outlined subsequently, involves solutions of five or six coupled,
nonlinear and multidimensional partial differential equations
and thus, involves a more complex computational task than those
96
reported earlier in Chapter 2. Nevertheless, the model having a
rigorous fundamental basis, is expected to provide relatively
more accurate description of the relevant phenomena associated
with the continuous casting operation^^' .
Thus to investigate various thermal phenomena (e.g., heat
flow and solidification) during continuous casting of steel, a
mathematical model based on the conjugate fluid flow and heat
transfer approach has been developed in the present study.
Towards this, relevant previous work, derivation of the governing
equations of fluid flow and thermal energy transport, numerical
solution procedure together with the adequacy of numerical
predictions with reference to the reported plant scale
measurements are described in the subsequent sections of the
text.
3.2 LITERATURE REVIEW
Although conjugate fluid flow - heat transfer model has a
sound fundamental basis, the approach has so far been relatively
less frequently adopted by the researchers investigating
mathematically heat transfer related phenomena in CC. According
to the present author, the inherent computational complexity
associated with the model (e.g. numerical solution of turbulent
Navier-Stokes equation together with thermal energy transport)
appears to be one of the principal reasons for such an
11 12
observation. Thus, so far, only few investigators ' have
reported studies based on the concept of conjugate fluid flow and
heat transfer. Remaining of the studies are either exclusively on
heat transfer^ (i.e*/ via artificial effective thermal
97
conductivity approach) or purely on fluid flow^^ in cc.
The very first study that took into account the possible
interactions between fluid flow and heat transfer in cc, has been
reported by Szekely and stank^^. in their study, the
investigators^^ adopted the general approach of Mizikar® (viz.,
effective thermal conductivity model) with a major modification
of assigning a definite flow pattern in the molten pool to deduce
the convective heat flow terms in the governing equation. The
investigators, considered the following three cases in their heat
flow calculations ;
(i) potential flow of fluid in the liquid pool was assumed
and the velocity field was deduced from the inlet velocity of
the pouring stream as source,
(ii) a high effective thermal conductivity in the liquid
pool was assumed to account for the convection and turbulence
within the pool, and
(iii) complete lateral mixing in the pool was assumed.
The results of computations for the three idealized flow
conditions mentioned above, were finally compared with each
other. It has been reported that the liquidus and solidus
profiles remained almost insensitive to the choice of any of the
three types of flow conditions. This illustrates an apparent
independence of solidification from the flow field in the liquid
pool. However, the rate of release of superheat was found to
depend markedly on the flow field in the molten pool. In the same
study, the investigators^^ also carried out a theoretical
analysis for the dispersion of tracer in the pool region and have
98
reported that the mixing and flow field within the pool play a
significant role in the floating out of the inclusion particles
from the melt during continuous casting operation.
The study by Szekely and Stank^^, although based on a
simplified approach of fluid flow analysis in CC, nevertheless,
represents an oversimplification of the actual process. Thus,
their model is not likely to be adequate enough to predict the
actual flow field and its associated influence on heat transfer
in CC.
In a subsequent study, Szekely and Yadoya^® developed a
model from a more fundamental considerations. They considered
turbulent flow in the molten pool and applied conservation
equation for mass, momentum, and energy, to predict the turbulent
flow field, temperature field and tracer dispersion respectively
in the upper pool region (i.e., the mould region). In addition,
they carried out extensive water model studies^^'^® and compared
the theoretical predictions of the velocity field with the
results of water model experiments.
3 6
Szekely and Yadoya essentially considered steady state,
two dimensional fluid flow and heat transfer in cartesian as well
3 6
as cylindrical polar coordinates. The investigators assumed
turbulent flow in the liquid pool and used the equivalent stream
function/vorticity - transport equations for computing the flow
field in the pool region. Furthermore, the spatial distribution
of turbulent viscosity was deduced from the Kolmogorov-Prandtl
mixing length model^^'^°. To avoid computational complexity,
these investigators^®, however, ignored the solidification
phenomena and its influence on the pool hydrodynamics. Thus,
99
there was no latent heat release term and instead, a viscous
dissipation term in addition to the standard convection and
turbulent diffusion terms, was considered in the governing
thermal energy balance equation. Finally, a general material
conservation equation was employed for predicting the
distribution of tracer concentration in the liquid pool region.
For a given set of casting conditions, the set of governing
partial differential equations were solved iteratively using the
finite difference numerical procedure, and embodying a prescribed
inlet conditions (viz . , known pouring temperature and inlet
velocity of stream) together with a prescribed heat flux at the
mould wall as boundary conditions. Finally, the predicted
velocity fields were compared with the results of water model
study and a reasonably good agreement (qualitative only) between
predictions and experimental measurements was demonstrated. The
predicted temperature fields and tracer concentration profiles
were also found to be quite consistent with those observed in the
3 6
water model study
Szekely and Yadoya , however, did not take into account the
influence of heat transfer (viz., solidification and mushy zone
formation) on fluid flow and consequently, the results reported
are somewhat oversimplified and hence to some extent physically
unrealistic. Nonetheless, the investigations provided the
necessary guidelines for further development of a more realistic
model. The effort of Szekely and Yadoya can be considered to be a
pioneer undertaking as this was the first detailed reported study
involving application of the theory of fluid flow and turbulent
transport to metallurgical systems, particularly in continuous
100
casting .
In a later
study, Asai
and
Szekely^^
made
further
improvements in
the model
reported by
the
earlier
OK O Q
investigators ' ,
by taking
into
account
the effect of
solidification (i.e., latent heat release) and the mushy zone
formation on the pool hydrodynamics and the resultant energy
transport in CC. The investigators^^ applied their model to
analyze fluid flow and heat transfer in various billet casters
and validated their model against the actual plant scale
33
measurements reported in literature
Asai and Szekely^^ considered a steady state two dimensional
fluid flow and heat transfer situation and accordingly, developed
a model in cylindrical polar coordinate system. For the velocity
field calculation, stream function and vorticity transport
equations were considered. Further, Kolmogorov-Prandtl mixing
length model^^'^®, with some modifications, was applied to deduce
the turbulent viscosity in the pool. Resistance to flow produced
by the solid matrix in the mushy zone was taken into account by
assuming its viscosity to be 20 times the molecular viscosity of
steel. In the computation of temperature field, latent heat of
solidification, released in the mushy zone, was estimated from
the corresponding change in the solid fraction assuming
ecpiilibrium solidification of steel. The governing fluid flow
equations together with energy balance equations were solved
using the finite difference procedure and for the two distinct
sets of input conditions considered. Velocity fields, spatial
distribution of turbulence kinetic energy and temperature fields
101
were predicted. Solidification profiles thus estimated from the
predicted temperature fields were compared with corresponding
experimental measurements reported in the literature^^ . Predicted
and the measured shell thicknesses were mostly found to be in
reasonable agreements with each other.
In addition to these, computed velocity fields and effective
viscosity were subsequently applied to calculate the transient
distribution of tracer concentration and trajectory of inclusion
particles within the liquid pool. Towards this, an appropriate
transient material conservation equation was considered in the
numerical calculation procedure. However, agreement between
theoretically estimated and experimentally measured mixing rates
were in general less satisfactory. In contrast to this, the
computed trajectory of the inclusion particles within the pool
was found to be quite consistent with the general expectations.
. 12
In a more recent study, Flint and coworkers developed a
steady state three dimensional conjugate fluid flow - heat
transfer model applicable to the mould region of a continuous
slab caster. These investigators assumed turbulent flow in the
liquid pool, and considered mass, momentum, and enthalpy balance
equations to compute the velocity, pressure, and temperature
fields. The turbulent properties in the liquid region were
estimated from the k-e model. Latent heat released during
solidification, and resistance to fluid flow in the mushy zone
were taken into account by incorporating appropriate source terms
in the governing fluid flow and heat transfer equations (e.g.
momentum sink and latent heat source terms respectively) . These
source terms were assumed to be dependent on the liquid fractions
102
in the mushy zone. Scheil's equation^ ^ was applied to calculate
the local liquid fractions in the mushy zone, and using the
numerical procedure proposed by Voller and Prakash'*^, above
mentioned source terms were estimated for the numerical
computations of velocity and temperature fields. These
investigators used TEACH-T code of Gosman and Ideriah"^^ to solve
the governing equations and associated boundary conditions.
12
They analyzed the influence of various numerical parameters on
the computed results, and reported that the numerical parameters
such as grid configuration, discretization technique, value of
turbulence model empirical constants, method of treating the wall
drag, and the input turbulence level, have significant effects on
the model predictions.
. 12
Flint and coworkers have also reported comparisons between
the 2D and 3D fluid flow - heat transfer models as well as the
effective thermal conductivity and fluid flow - heat transfer
models. Two dimensional model was found to give poor quantitative
predictions of fluid flow and flow-induced shell growth in the
mould in comparison with the 3D model. Similarly, the effective
thermal conductivity model was found to be less satisfactory than
fluid flow heat transfer model.
Influence of some of the operating parameters of slab
casting such as, superheat of liquid steel, casting speed, and
submerged entry nozzle (SEN) , on the predicted velocity and
12
temperature fields were also studied by Flint and coworkers
Finally, the model predictions were validated against the
experimental observations made on a full-scale water model.
103
3.3 FORMULATION OF TRANSPORT EQUATIONS FOR THE PRESENT STUDY
3.3.1 Assumptions in Modelling
In deriving the governing fluid flow and heat
transfer equations, the following assumptions were incorporated
in the mathematical model.
(a) Momentum and heat transfer have been assumed to be
essentially a steady state, two dimensional phenomena with
reference to an axisymmetric continuous casting operation. Thus,
for cartesian coordinate system, fluid flow and heat transfer
were analyzed only in the central vertical plane of the cast
section. Also, due to the symmetry of the configuration chosen,
velocity and temperature fields were computed only in one half
section of the chosen central vertical plane.
(b) While considering the mathematical simulation in
cartesian - coordinate system, influence of one of the two
transverse directions (i.e. y-direction) on the predicted
velocity and temperature fields were neglected (i.e. no influence
of corner region) . Thus, for continuous casting of rectangular
cross-sections (e.g. billets) , phenomena at the mid face and the
corresponding central vertical plane has been taken into account.
Similarly, for cylindrical polar coordinates (e.g- round
billets) , influence of 0 - direction was ignored.
(c) Solidification front was assumed to be planer with
respect to the adjacent liquid. Also, the influence of movement
of solidification front on flow field in the pool was ignored as
a first approximation.
(d) Flow in the liquid pool has been assumed to be induced
by the momentum of the incoming pouring stream as well as by the
104
thermal buoyancy (i.e. natural convection phenomena). Influence
of other secondary factors such as bulging, suction due to
solidification shrinkage etc., were ignored.
(e) Only the perfectly vertical cast section has been
considered as the calculation domain (i.e. no influence of strand
bending or the curvature effect) for necessary mathematical
simulation.
(f) Effect of mould oscillation has been ignored.
In addition to these, most of the assumptions made during
the study on artificial effective thermal conductivity approach
(viz.. Chapter 2, Section 2.3.1) were also made applicable to the
present study. Therefore, these are not reproduced here.
3.3.2 Governing Equation of Fluid Flow Within the Liquid
Pool and Boundary Conditions
The flow field within the liquid pool of a
solidifying section must satisfy the mass conservation (i.e.
continuity) and the momentum conservation equations. The
governing fluid flow equations, therefore, for a two dimensional,
steady state, incompressible, turbulent flow may be expressed by
the following general partial differential equations.
Equation of continuity,
= 0
au
az
■ s ixH =
Equation of motion in axial direction.
... (3.1)
a
azi
f aul
^ 1
a
au)
l^ef f az J
+ -
ax [“
*^effaxj
u
. . (3.2)
105
where, axial momentum source term per unit volume is,
a — ^ fii J. i ^ r
- azPeffazJ + a Sx[“ "effSzJ ' - V
. . . (3.3)
Equation of motion in transverse/radial direction,
ll(H + I lx(“H " - li + Iz^ff H) 3 lx(“ “eff S)
where, radial momentum source term per unit volume is,
®v ^ az(*^eff a *^eff ai]
. . . (3.5)
The parameter 'a' appearing in Eqs.(3.1) through (3.5), is a
scaling factor (e.g. to be visualized as an index of coordinate
system) , which is unity for the cartesian coordinate (applied to
the analysis of square billets) , and (a = x = r) for the
cylindrical polar coordinate (for analysis of round billet)
systems. Furthennore , for the Cartesian coordinate system, the
quantity (^i^^^v/x ) in Eq. (3.5) is set equal to zero.
In Eqs.(3.1) through (3.5), is the effective viscosity
which is defined as the sum of the molecular and turbulent
contributions to the viscosity, i.e.,
= h + ... (3.6)
The turbulent viscosity (/x^) in Eq.(3.6) is not a physical
property of the fluid and instead, is largely determined by the
nature of flow.
In Eq.(3.3), the source term (S^) in the axial momentum
balance equation contains ["Pg/3(T-T^) ] term. This takes into
106
I. Flow driven by forced convection
II. Flow driven by free convection
III. No flow or stagnant region
Fig. 3.1. Schematic of the flow pattern in the liquid
pool of a continuously cast billet.
107
accoun't 'tha ln£luenc@ of buoyancy force (i.e. Bousslnesc} source
13
’term ) , generated by the difference in density due to difference
in temperature (i.e. natural convection) within the liquid pool
of a solidifying casting (Fig. 3.1). In none of the earlier
studies on continuous casting, influence of buoyancy on the flow
of liquid steel within the pool has been been taken into account.
However, as the literature indicates^^'^^, flow in the lower pool
region is expected to be influenced by the buoyancy and in view
of these, thermal buoyancy has been included in the momentum
balance equation along the axial direction in the present study.
Inside the liquid pool there are temperature gradients along
the axial as well as the radial directions. Due to this, the
density of liquid steel increases gradually as it moves form the
hotter to the cooler region within the liquid, pool, and
therefore, have more and more tendencies to settle down during
its descent through the pool. Therefore, the buoyancy force in
the present investigation has been considered to act in the same
direction as that of gravity (i.e. axial direction) .
Boundary Conditions:
The boundary conditions assumed for the momentum balance
equations (3.1 - 3.5) are:
(a) At the meniscus (Z = 0) ,
(i) inside the pouring stream,
0 s X s r^, u = and
(ii) outside the pouring stream,
r^ < X s R, u = 0 and
V = 0 ...(3.7)
av
dz
* 0 ...(3.8)
108
(b)
At the exit
or
outflow boundary (Z =
L),
VI
X
VI
o
3u _ av
az ° az ~
0
(c)
At the axis
of
symmetry (X = 0) ,
0 s Z ss L
9u - , 3v
ax ° ax ~
0
...(3-10)
(d
At the side
wall (i.e., mould wall)
(X=R) ,
VI
tSJ
VI
o
u = U.^ and V =
in
0
A schematic diagram of calculation domain and the relevant
boundary conditions are shown in Fig. 3. 2. The inlet velocity of
the pouring stream (Eg. (3.7)) was derived from the global
continuity, setting (cross sectional area of nozzle x inlet
■f'
velocity of the stream) equal to (cross sectional area o
mould X casting speed) . At the meniscus and the axis of symmetry,
no normal velocity components may exist. Therefore, these were
set equal to zero at these locations. Similarly, At all the solid
surfaces (i.e., mould wall and solidification front) fluid motion
* s I^ul^
ceases to zero. Further, the entire domain moves wirn
motion (U ) . Therefore, at the mould wall, the solidif
° axial
front as well as in the completely solidified region, the a
velocity component (u) was set equal to the casting
whereas, the corresponding radial velocity component wa
zero values. The exit or outflow boundary was assumed to
located far away from the meniscus (preferably in a complete y
- nut flow
solidified region) so as to keep the influence or
boundary on the predicted upstream flow field to a minimum. Thus,
the gradients of both axial and radial velocity componen
set equal to zero at the chosen exit boundary. The selection
an appropriate outflow boundary and the associated compu
109
Fig. 3.2, Schematic representation of the 2-D calculation
domain and the boundary conditions applied
in the computation of velocity and temperature
fields.
110
implications of the same are discussed in detail in a later
section.
3.3.3 Modelling of Turbulence Within the Liquid Pool
The diffusion terms embodying effective viscosity
^*^eff^ momentum balance equations (Eqs. (3.2)-(3.5) ) , takes
into account the influence of turbulent and viscous dissipation
of momentum within the liquid pool. As mentioned already/ the
effective viscosity which is the sum of molecular and turbulent
viscosity (i.e. = U + r is strongly position-dependent,
and is largely determined by the nature of hydrodynamics and
turbulence present in the system. Various types of turbulence
models are available in literature^^'^^ for estimating the
turbulent viscosity. For example, the well known K - c model^^ is
generally accepted to give a proper representation of turbulence
in the high Reynolds number recirculatory flow systems.
Similarly, as pointed out already, the Kolmogorov-Prandtl mixing
length model can also be applied to derive the turbulent
viscosity in the liquid region^^. Towards this, it is important
to note here, that theoretical as well as water model studies on
CC have revealed that within the upper pool region only, the flow
is predominantly turbulent and recirculatory ' . In contrast,
turbulence is practically insignificant in the lower pool region
(Fig. 3.1). Therefore, in the computational scheme, it would be
more realistic to apply a turbulence model to the upper pool
region only and practically consider laminar flow conditions in
the lower submould region. However, as is well known, it is
difficult to apply the K - e or in fact, any other differential
Ill
turbulence model (one equation or two equation models)
selectively in specific regions due to the following constraints.
(i) Outflow boundary is not known a-priori. Therefore,
specification of boundary conditions on the turbulence parameters
(say K and e) to any arbitrarily chosen outflow boundary is
difficult,
(ii) Since nature of the flow is considerably different
in the upper and lower pool regions if we consider turbulent flow
to be confined to the mould region only, it is difficult even
then to apply any first or higher order turbulence model
selectively in the mould region due to the imposed uncertainty in
the value of turbulence parameter at the mould exit (no boundary
condition can in principle be applied on the dependent variables
at the mould exit since the latter truly is not a physical
outflow boundary) .
(iii) At the solidification front and in the mushy zone, the
governing turbulence transport equations (say K and e equations)
will require considerable modifications. However, presently,
nothing is known on this issue.
Consequently, considering the complexity of the pool
hydrodynamics and the associated nonlinearity of the equations
dealing with conjugate fluid and heat flow, it was decided to
evaluate the turbulent properties in the liquid pool using some
simplified model rather than a more rigorous and advanced
differential model of turbulence (e.g., the K - e model). In this
46 47
context, applicability of the Pun - Spalding formula ' of
average effective viscosity to the present situation was
analyzed critically. The Pun - Spalding formula is based on the
112
Prandtl mixing length model of turbulence and was originally
developed for the analysis of flow in the centrally fired
axisymmetric combustion chambers^^. in principle the Pun-Spalding
formula can also be applied to the other equivalent systems
involving flows dominated by inertia force, in sudden expansion
typs geometries. The formula, though not quite conceptually
applicable, has been applied successfully to the study of inertia
dominated flows in the gas stirred ladle systems^® '
In CC, discharge of liquid steel from the pouring nozzle to
the mould is conceptually analogous to the flow through a sudden
expansion, such as the one considered by Pun and Spalding^® '
This follows since the flow particularly, in the mould region, is
dominated by inertia force (e.g., entry Reynolds number at
discharge nozzle varies typically between 5x10^ to 9x10^) .
Therefore, as a first approximation, average effective viscosity
was calculated for the mould region only using the Pun - Spalding
46 47
formula ' , which in the present investigation can be
represented in the following form:
U
eff
= A Z- p2/3 o. )
m m ^ ^ in'
(3.12)
In Eq. (3.13), A is a dimensionless empirical constant (=
0.012)^^'^^^, and m is the mass flow rate of steel and is
estimated via the following expression:
p U.
^ in
. . . (3.13)
In the submould region, somewhat less turbulence has been
113
assumed and hence, the effective viscosity in this zone was
arbitrarily set to only about 30 - 50 pet. of the mould effective
viscosity values. The sensitivity of the predicted flow and
temperature fields to the choice of the effective viscosity value
will be addressed rigorously in Section 3.5.2.
3.3.4 Governing Equation of Heat Flow and Boundary
Conditions
The governing equation of heat flow expresses the
conservation of thermal energy over a volume element within the
system. For a steady state, two dimensional heat flow situation
during continuous casting, the appropriate thermal energy balance
equation can be represented by the following general expression:
IzHo’’] + +i|((«pcvT] = 3|(r^„i)
3 3x(“ ^effsx) ® ...(3.14)
The first term on the L.H.S. of Eg. (3.14) represents heat
flow in the axial (withdrawal) direction (X) due to the bulk
motion (U^) of the descending strand. Whereas, the second and the
third terms are the convection terms due to velocity components u
and V respectively. On the R.H.S., the first and the second term
represents conduction of heat along the axial (X) and the radial
(Y) directions respectively, and S is the latent heat source term
defined as:
'at '
s
(3.15)
similarly/ the effective thermal conductivity in Eq. (3.14) is
defined as:
r
eff
K +
. . . (3.16)
Furthermore/ in Eq.(3.16) is the turbulent Prandtl number and
is defined as :
=
= turbulent thermal diffusivitv
t turbulent nioinentuin diffusivity
... (3.17)
For most of the turbulent flows, x* However, a typical
value of 0.7 for liquid steel, has been used by Asai and
11
Szekely in their studies. Assuming cr.“ 1 i.e. / a. = v. / it is
readily seen that:
or,
Since
Therefore/
K^/pC =. u^/p
"eff “ "t
■’eff = ■' * "eff=
. . . (3.18)
It may be noted that the energy balance equation is
applicable to each zone in continuous casting, viz. liquid/ mushy
zone, and the solidified region. Furthermore, the derivative
(df^dZ) appearing in Eq. (3.15) is by definition zero everywhere
except in the mushy zone viz., ( . In the solidified
region the velocity components (u and v) become zero and thus, a
pure conduction like equation results for the solidified region
from Eq. (3.14), with additional contribution due to bulk motion
(or casting speed) of the descending strand. However, in the
115
liquid pool, heat is transferred by convection (i.e due to fluid
velocity components u and v) and turbulent conduction only, and
the bulk motion contribution in the liquid core is zero.
Therefore, in the regions T > the energy balance equation
(Eq.(3.14)) becomes analogous to the momentum balance equations
viz. Eqs. (3.2) and (3.4).
In the present investigation, latent heat released during
solidification has been evaluated from the solid fractions in the
mushy zone assuming equilibrium solidification of steel.
Procedure for the estimation of latent heat release from the
relevant equilibrium phase diagram (Fig. 2. 2) has already been
described in Section 2.3.4, and is therefore not reproduced here.
The boundary conditions applied to Eg. (3.14) were also
exactly identical to those considered for the effective thermal
conductivity model. For clarity of presentation, temperature as
well as the velocity boundary conditions are shown in Fig. 3. 2,
and are also stated below.
(a) At the meniscus (Z = 0),
(i) inside the pouring stream,
0 s X s r , T = T ... (3.19)
0 0
(ii) outside the pouring stream,
r^ < X s R, 3T/az = 0 or = 0 ...(3.20)
(b) At the exit boundary (Z = L) ,
0 s X R, dl/dZ = 0 ...(3.21)
(c) At the axis of symmetry (X = 0) ,
0 s Z s L,
dT/dX = 0
. . . (3.22)
117
differential equations and the associated boundary conditions to
their equivalent dimensionless forms prior to carrying out any
detailed computations. As is well known, in the dimensionless
form the system of equations become essentially scale free, and
hence the chances of accumulated error in the predicted result
are less. In addition to these, the orders of magnitude of
various terms in the non-dimensional partial differential
equations become comparable to one other and therefore. Influence
of any particular parameter on the computed results can be
conveniently estimated/studied.
In the nondimensionalization procedure, the following
dimensionless variables were defined.
(i) dimensionless space variables
(ii)
(iii)
Z* = Z/R
X* = X/R
aimensionless velocity components
0
u
- “/Of
in
V -
dimensionless pressure variable
p' - p/ipvf 1
(iv) dimensionless temperature variable
and finally, (v)
T* = T/T^
the relevant dimensionless groups.
Reynolds Number (Re) = P^in^/^^eff
Peclet Number (Pe) = P^in/(^Qff/^)
Thus, the fluid flow equations and the associated boundary
conditions in dimensionless form (say in cartesian coordinate
118
system) can be conveniently presented as:
Equation of continuity
+ ^ . 0
az ax
. . . (3.26)
Equation of motion in axial direction
a
az'
f ♦
u u
* ‘I
U V
a
*
ri
Re
au
«
j
ax
k J
az
az
k.
az J
ax*
(k. ^
Re *
ax
+ s.
u
where, axial momentum source term per unit volume
a
az'
fi
au )
+ L-
fi
av*]
Re
az*J
«
ax
Re
k
az*
. . . (3.27)
- Rg|3T^(T*- 1)/U^
in
. . . (3.28)
Equation of motion in transverse/radial direction
a
az'
( • *1
U V
V V
m
a
•
Ik-
Re
•
1 J
ax
ax
az
az J
£_ [k- £5L
ax*r® ax*
+ s.
. . . (3.29)
Where, transverse/radial direction momentum source term per unit
volume
* d
ri
au
a
ri
av
Re
ax*^
+ — J
ax
Re
ax*.
. . . (3.30)
The corresponding boundary conditions applied to the
momentum balance equations in dimensionless form are:
(a) At the meniscus (Z =0)
(i) inside the pouring stream
0 s X* s r , u 1 and v — 0
. . . (3.31)
119
(ii) outside the pouring stream
u=o and av*/az* = o
(3.32)
(b) At the exit boundary (z* = l*)
0 s X s 1, au /az = 0 and av*/az*= o ...(3.33)
(c) At the axis of symmetry (X* = O)
O^ZsL, au/ax = 0 and V* = 0
(d) At the side wall (i.e., mould wall) (x*= 1)
0 s z*s L*, u* = U
and V = 0
. . (3.34)
. . . (3.35)
Similarly, the energy balance eguation and associated
boundary conditions in the dimensionless form become as follows:
T-) ^ ^(u-T-) f ^(vY)
az az ax
a (1 dT
* Pe *
az az
+ + s’
ax* ax*
y
. . . (3.36)
In Eq.(3.36), S is the dimensionless latent heat source term,
and is defined as:
U* AH.
0 f
(atv az')
(3.37)
The corresponding dimensionless temperature boundary
conditions are:
(a) At the meniscus (Z* - 0)
(i) inside the pouring stream
0 s x*s r*.
• ♦
T
. . . (3.38)
120
(ii) outside the pouring stream
r* < X*S 1 , dT*/az* = 0 ...(3.39)
(b) At the exit boundary (Z* = L*)
0 S X* S 1, aT*/dZ* = 0 ...(3.40)
(c) At the axis of symmetry (X* = 0)
0 S z* S L*, dT /ax* = 0 ...(3.41)
(d) At the cast surface (X* = 1)
(i) mould region
■0 * - »-33/z/ D„) 10®
. . . (3.42)
(ii) secondary cooling zone,
^m ^ ^ " ^s' % ~ pctjrC^s ” ^w)
.. (3.43)
(iii) radiation cooling zone
Lg < Z*:s L*, q* = (o-eej /pCU.^^Tj (©*'
C)
(3.44)
in which.
0 = T + 273
O O
3.3.6 Modelling of Fluid Flow in the Mushy Zone
Computation of velocity field within the mushy
zone is much more complex than in the bulk liquid steel, as the
mushy zone typically involves flow through complex interdendritic
channels. Furthermore, flow is also induced in the mushy zone by
suction caused by the solidification shrinkage. In addition,
rejection of solute elements by the solidifying dendrites into
121
the interdendritio spaces (i.e. eicrosegregation phenomena)
poses additional complications.
To take into account the exact influence of all these
factors in the theoretical analysis of fluid flow and heat
transfer in the liquid pool, considerable difficulties
(fundamental as well as computational) were anticipated.
Consequently, an ad— hoc simplified approach has been considered
to model the fluid flow in the mushy zone. Towards this, it is
important to mention here that Asai and Szekely^^ considered the
increased resistance to flow produced by the solid matrix in the
mushy zone by increasinq the liquid steel viscosity by a factor
of 20 in the mushy zone. The investigators^^ derived this factor
from a typical viscosity-temperature correlation reported in the
literature . In the present study, the same assumption was made
for simulating the possible influence of mushy zone on the
resultant fluid flow. However, as will be described
subsequently, influence of other choices of mushy zone viscosity
on the predicted flow and temperature fields were also studied
computationally.
3.3.7 Choice of the Outflow Boundary
Typically, the axial gradients of the dependent
variables are assumed to be zero (Eq. (3.21)) at the exit plane
(e.g. the outflow boundary) , at which a fully developed flow
situation must truly exist. Thus, with such a constraint at the
outflow boundary, it was observed that no meaningful solution
(i.e. realistic velocity field) could be obtained if the outflow
boundary is placed anywhere arbitrarily. It was also found that
122
the erroneous/unrealistic positioning of the outflow boundary
leads to uncertainty in the boundary conditions at the exit plane
and consequently, in turn produces unrealistic prediction.
In billet casters, pool depth often extends much below the
cast strand, and flow reversal is significant even in the lower
portion of the liquid pool. Hence, the imposed zero gradient
boundary conditions at any arbitrarily chosen outflow boundary
may not be a realistic one, unless the chosen exit plane lies in
the completely solidified region. Thus, due care was taken in all
computations to ensure that exit plane was located far enough
downstream in the completely solidified region, so as not to
influence the upstream results. Thus, in the computations
reported subsequently, the exit plane was considered to be
located at 10 m downstream of the inlet plane.
3.4 NUMERICAL SOLUTION OF THE GOVERNING PARTIAL DIFFERENTIAL
EQUATIONS
3.4.1 Numerical Solution Procedure
Numerical solution of the governing fluid flow and
heat transfer equations were carried out by the popular TEACH-T
computer code, originally developed by the Computational Fluid
Dynamics research group at the Imperial College, London. However,
as will be discussed below, the original TEACH-T code was
considerably modified before it was used in the present numerical
investigation .
The TEACH-T code is based on the control volume based finite
difference numerical procedure and incorporates formulations
123
based on the primitive variables (u,v, and p) , instead of the
retrieved stream function and vorticity^^^Sl^
implicit Method for Pressure-Linked Equations) algorithm,
originally developed by Patankar and Spalding^°, is applied to
numerically solve the pressure-velocity coupling in the governing
fluid flow eq[uations. The computer code is suitable for
axisymmetric, two dimensional, turbulent or laminar recirculating
flow calculations with variable thermophysical properties.
Furthermore, it can solve the set of governing partial
differential equations in Cartesian or cylindrical polar
coordinate systems. The K-e model of turbulence has been embodied
in the code for the computation of turbulence parameters. The
overall structure of the code is modular, thus enabling any
dependent variables to be removed or added at will.
For the flow field calculations staggered grids were
employed for its special attributes^®' where, the scalar
variables (i.e. P and T) were specified at the central nodal
point of the control volumes, and the vectors (i.e. u and v)
were located on the cell interfaces (as opposed to centers) .
Figs. (3.3) and (3.4) present schematic of grid distribution and
various types of control volumes respectively, considered in the
present computational work. The relevant governing equations were
integrated over their respective control volumes to yield a
system of discretization or finite difference equations. The
procedures of discretization of the governing partial
differential equation has already been described in detail in
chapter 2 of the present work, and therefore, are not reproduced
here, in essence a general variable (p (say u,v, or T) at any
124
Fig. 3.3. Schematic of the grid layout and control
volumes for vector (u & v) and scalar
(P & T) variables.
(a) (b) (c)
Control volume for Control volume for Control volume for
scalar variables (i.e P ^ T) oxial velocity component (U) radial velocity component
Fig. 3.4. Schematic of three typical control volumes for scalar (i.e. P & T)
and vector (u i v) variables employed in the numerical
computation scheme.
126
nodal point P (Fig. 3. 4) can be represented in tenns of its
neighbour point coefficients (E, W, N and S) via the following
discretization equation:
Ap^p = Ag<^j, + + Ag^g + Sy ...(3.45)
in which, Ap = ^ “ ®P
In Eq. (3.45), A's are the coefficients of discretization
equation embodying the combined influence of both convection and
diffusion contributions to <f> by the neighbouring grid points.
It
while S stands for the 'source term' as defined in Eqs. (3.2) and
(3.4). Again, linearization of S can be accomplished, as in
Eq. (3.45) by using the variable <p as follows:
S = Sy + Sp^p ...(3.46)
In Eq. (3.46), Sy and Sp are the two functions which depend on the
particular (p variable concerned.
During the discretization procedure, transport properties
(e.g. viscosity and conductivity) at the control volume faces
were estimated via a more accurate harmonic mean interpolation
method^*^'^^. However, in computational fluid dynamics, a
realistic representation of the convection and diffusion terms is
essential to the accuracy and convergence or even stability of
the iterative calculation scheme. Therefore, the hybrid
difference scheme^^ was incorporated in the computation scheme
for representing the combined convective and diffusive
contribution of momentum and heat from the neighboring control
volumes to the central ones. The set of discretization equations
solved iteratively by the well known Tri-Diagonal Matrix
were
127
Algorithm (TDMA) incorporating an efficient line by line solution
procedure . To this end, routine underrelaxation practice to the
dependent variables was applied. The underrelaxation practice
applied together with the scope of convergence/convergence
criterion adopted in the present study are discussed in the
relevant subsequent section.
3.4.2 Numerical Procedure for Incorporating the
Influence of Solidifying Shell on Fluid Flow and
Heat Transfer
The velocity components (i.e., u and v) are zero
at the solidification front, while the entire domain is moving
with casting speed (U^) . Therefore, in the numerical solution
procedure applied to the fluid flow equations, the axial velocity
component u was set equal to the casting speed and the radial
velocity component v was set equal to zero at the solidification
front as well as in the entire solidified region. Mathematically,
these conditions on the governing fluid flow equations can be
expressed as,
U = U„, v = 0 ...(3.47)
The restrictions imposed by Eq. 3.47 provides a realisti
description of bulk motion of the descending strand with respect
to the fluid flow in the molten pool in continuous casting.
The numerical procedure for prescribing these conditions,
within the solidified region as well as at the solidification
front, therefore, should be such that the solution procedure is
able to reflect the exact prescribed values of velocity
128
components in these regions. To this end, there are two numerical
techniques available®^, 30^ first^O involves
artificially assigning a very high values (say 10^°) of viscosity
in the solidified region together with the desired values at the
solidification front. This procedure, however, rests on the
ability of the numerical procedure to handle a large step change
in the values of transport coefficient (e.g. effective viscosity
etc.). In this context, harmonic mean interpolation for
estimation of transport coefficients at the cell interfaces has
been found to be the most appropriate^^. The other procedure^^,
called the 'cell porosity^ or 'blockage ratio' technique,
involves blocking off preferentially those control volumes lying
in the inactive zone (i.e. solidified region) so that only the
remaining control volumes form the active domain for the
computation of flow field. In the present study, the latter
procedure has been employed to take into account the influence of
the growing shell on the fluid flow and thus on the turbulent
convective transport of heat. It is to be emphasized here that
both these techniques are expected to provide identical estimates
of flow parameters in the calculation domain.
In the cell porosity method, for each control volume face, a
blockage ratio (from 0 to 1) was defined. Thus, for the control
volume face lying fully in the solidified region (i.e.
completely blocked to flow) , blockage ratio was defined as unity.
Similarly, for the face lying completely in the liquid region,
blockage ratio was set equal to zero. Those control volume faces
which are cut by the solidification front (i.e. only partially
in the solidified region) , blockage ratios were estimated from
129
the fraction of the area of control volume face blocked by the
solid. During each iteration, position of solidification front
for each axial station was derived from the predicted temperature
field and a smooth curve fitted through the loci of the
solidification front. Subsequent to this, the positions of
intersection of the fitted curve (e.g., numerical solidification
front profile) and all the relevant control volume faces were
computed. From these, blockage ratios of the various faces of
control volume were calculated. Intersection of a radial velocity
control volume and the solidification front together with the
procedure used for the evaluation of the blockage ratio is
illustrated schematically in Fig. 3. 5.
Subsequent to the calculation of blockage ratio, the
original coefficients of the discretization equation (Eq. (3.45)),
porosity) for the four faces of any two-dimensional control
volume .
When all the blockage ratios for a control volume were 1
(i.e. in the completely solidified region) , all the neighbouring
coefficients of the discretization equation became zero, and
hence, the grid node became completely isolated from its
neighbors. The value of a variable ^ at such a node could then be
130
LIQUID REGION Z(I)
(I.J
Portially blocked faces:
XsL w
foceV BRw»1-
Xc, .-X(J 1)
face 'e* BRe * 1 ^~^X
* Deduced via curve fitting through
0 set of solidus temperatures
Fig. 3.5. Schematic of a typical radial velocity control volume
located in the vicinity of the solidification front and
evaluation of blockage ratios for various control
volume faces.
131
fixed at any desired value, 6^ ^ . (f^-r tt - tt
' ^P, desired example, U = and
V = 0 at the nodes just on the solidification front or in the
solidified region for the present case) by redefining the
components of the source term in the discretization equation as :
Sy = 10^° X 0
P, desired
and
Sp = - 10
30
. . . (3.49)
With such a prescription, Eq. (3.45) reduces to
• ®P *P “ 1
►
or, <f>-p - - ^u/^P “ ^P, desired
. . . (3.50)
Such a technique allowed the value of the dependent variable
to be fixed whereever needed. Thus, during each iteration, the
velocity equations were solved by assigning u=U^ and v = 0 in the
completely solidified region. Subsequently, the temperature
equation was solved, in which the velocity components u and v
were both set equal to zero in the solidified region in order to
eliminate completely the thermal convection terms in the
governing heat flow equation, similarly, in the solid region,
true value of thermal conductivity of steel was applied when
solving for the temperature field.
3.4.3 The Computer Program
As mentioned already, the TEACH-T computer code
was employed for the computation of flow field in the present
study. However, the original computer code was modified
132
extensively with the addition of few more subroutines and several
other features for numerical solution of governing fluid flow and
heat transfer equations. The following modifications have been
incorporated in the original TEACH-T code:
(1) The TEACH-T code is capable of computing only the
laminar or turbulent flow fields. Thus, for the temperature field
calculations, a separate subroutine was developed and
incorporated in the code so that convective turbulent heat
transfer problem can also be solved numerically.
(2) For the calculation of solid fraction distribution in
the mushy zone and the associated rate of latent heat rel<£?.se, a
separate subroutine has been developed and added to the computer
program.
(3) A separate subroutine for the computation of pool
profile has also been developed and incorporated.
(4) A numerical procedure for estimating the blockage
ratios has been incorporated via a separate subroutine.
(5) The average effective viscosity formula of Pun and
. 46
Spaldxng has been applied as an alternative to the K-e
turbulence model.
In addition to these, for the implementation of boundary
conditions, the original subroutine PROMOD (which contains all
relevant boundary conditions) has also been modified considerably
in accordance with the specification of the present problem.
Also, a separate module for the relevant temperature boundary
conditions was incorporated in the PROMOD subroutine.
All the four new subroutines mentioned above, were
separately developed and tested against standard input data.
133
before incorporating into the original TEACH“T program. Detailed
flow chart of the modified TEACH code applied to the present
investigation is illustrated in Fig. 3. 6. Prior to initiating the
actual computations for the data set presented in Table 3.1/ both
uniform as well as non uniform grids of different configuration
were tried in an attempt to establish practical grid independent
solutions. Towards this, for a typical cylindrical billet^°, 200
X 18 grids and for square billets^^ 200 x 24 grids {Table 3.1),
were found to give satisfactory results. This corresponds to grid
spacing of approximately 50 mm and 3.5 mm in the axial and
transverse directions respectively. All computations were
performed on mini-super CONVEX computer available at I.I.T.
Kanpur. A convergence criterion of maximum normalized residual (=
5 X lO”^) was set on all variables, which for a general variable
(p is defined mathematically as ;
Residual = Ap^p - (Ag0g + + Ag^g + Sy)
Normalized Residual = ^(Residual) j/ (Total input momentum/heat)
. . . (3.51)
Each computation was carried out till the absolute sum of
residuals on u,v, mass continuity, and T all fell below their
stipulated values (i.e. the prescribed maximum normalized
residual value) . The data applied to the numerical computations
are presented in Tables 3.1 and 3.2 respectively.
134
135
3.5 RESULTS AND DISCUSSION
3.5.1 Some Considerations on the Scope of Convergence of
a Multidimensional Coupled Fluid Flow Heat
Transfer Problem
As mentioned already and illustrated in the
preceding sections, the fluid flow and heat transfer equations
describing the present problem are highly nonlinear in nature.
The source terms, some of the boundary conditions, together with
variable thermo-physical properties are in general, seen to
contribute to the nonlinearity. In addition to these, mutual
coupling between fluid flow and heat transfer equations through
the buoyancy term in the axial momentum equation and the presence
of solidified shell in the calculation domain lead to some severe
nonlinearity in the present problem. As a result of these
inherent complexities, numerical solution of the set of partial
differential equations presented considerable difficulties in
arriving at a converged solution.
Initially, several trial executions were carried out with
original (i.e. dimensional) forms of the momentum and energy
balance equations, and these failed to produce any meaningful
solution. Subsequently, it was observed that non-
dimensional izat ion of the governing eqpaations somewhat enhanced
the scope of arriving at the converged solution. Through
extensive computational trials it was further observed that
convergence could be obtained only for a narrow range of value of
the relaxation parameters (on u, v, and T respectively) . This
latter parameter and hence convergence was found to be sensitive
136
to the particular set of input conditions as well as the grid
configurations applied. On the basis of computational trials made
during the initial stage of the study, the following observations
were made:
(i) Underrelaxation of the dependent variables was
important, and the choice of values of underrelaxation parameter
was found to be critical from the view point of arriving at the
converged solution.
(ii) The value of underrelaxation parameter on a given
dependent variable was not unique but was a function of grid
layout and hence, for any given set of grid configuration chosen,
it was to be determiined through numerical trial and error.
(iil) The maximum number of iteration (or the computational
work) that was required for arriving at converged solution was a
function of both grid layout and the value of relaxation
parameter applied.
Thus, for each individual problem that is to be solved, a
large number of initial numerical trials have to be conducted.
This, according to the present investigator, appears to be a
major limitation in the application of turbulent fluid - heat
flow concept to the continuous casting of steel.
Intermediate results obtained during computations indicated
that the magnitude of dependent variables fluctuated over a wide
range before reaching the final converged solution. The extent of
fluctuations in the dependent variables (i.e., dimensionless axial
velocity component and temperature) with the progress
iteration is shown in Figs. 3.7 and 3.8 respectively. These
Clearly demonstrate the extent of the associated complexities
Dimensionless axial velocity,
137
Fig. 3.7: Variotion of the dimensionless axial velocity component _
at a monitoring location (i.e. r=0.005 m, Z=0.1 m) with
the progress of iteration.
Dimensionless temperatu
Fig. 3.8: Variation of the dimensionless temperature at a
monitoring location (i.e., r=0.005 m, Z=0.1 m)
with the progress of iteration.
139
involved in the numerical solution ot the present problem which
is essentially due to the coupling ot fluid flow, heat transfer,
solidification phenomena.
3.5.2 Sensitivity of Computations to the Choice of
Effective Viscosity Value
The Reynolds number at the liquid steel inlet/
nozzle was estimated to be of the order of 10^ or qreater.
Therefore, the flow in the liquid pool, particularly in the mould
region, can be safely considered to be turbulent. To this end,
any appropriate turbulence model can be applied in order to
estimate the required turbulent properties within the system.
However, estimation of effective viscosity or as a matter of fact
any other turbulence parameter in the liquid pool using a
rigorous turbulence model (e.g. the K - c model) may not be
appropriate for the present problem due to the reasons described
already in section (3.3.3). Therefore, as mentioned before, the
46
Pun - Spalding formula (Eg. (3.12) was applied to estimate the
average effective viscosity in the liquid pool for the mould
region only.
The expression in Eg. (3.12) indicates that increases
2
with increase in the rate of kinetic energy (i.e. conveyed
to the liquid pool by the incoming pouring stream. Also,
increases with increase in the mould diameter (Djjj) ^nd decreases
with any increase in the mould length (Lj^^) . Theoretical as well
as experimental studies^^'^®"^^ on CC indicate that turbulent
flow is confined only in the upper liquid pool region of the
solidifying casting. Consequently, in the present investigation.
Shell thickness, mnn
140
141
the Pun-Spalding formula^^ was applied to the mould region and
thus, effective viscosity was estimated from the dimensions of
the mould assuming negligible distortion in the pool geometry
(i.e., in the pool diameter) by the thickness of the solidified
shell. Furthermore, in the submould region arbitrarily, somewhat
less turbulence was assumed and towards this, approximately 50 “
70 pet. lower effective viscosity value than those estimated via
Eg. (3.13) for the mould was prescribed in the submould region.
It is to re~emphasized here that no differential model of
turbulence can be so conveniently applied over the entire liguid
pool, as it is possible with a bulk average turbulence model
(Eg. (3.12) ) .
Thus, the influence of prescribing the same l^^ff value
throughout the pool and a 50 pet. reduced value in the
submould region, on the numerically predicted shell thickness is
shown in the Fig. 3. 9. This clearly indicates that the assumption
of different turbulence levels in the submould region does not
affect the overall heat transfer rates significantly. The present
analysis thus revealed that the exact modelling of turbulence
phenomena is relatively less critical for predicting the
temperature fields and solidification phenomena in CC.
3.5.3 Modelling of Flow in the Mushy Zone and Its
Influence on the Computed Results
For the temperature field calculations, transfer
of heat in the mushy zone has been considered to take place by
the convection and the molecular conduction. Whereas, for the
velocity field calculation, resistance to the bulk flow of fluid
142
imparted by the solidifying dendrites in the mushy zone was taken
into account by artificially increasing the viscosity of the
mushy zone. As described already, Asai and Szekely^^ assigned a
value of viscosity in the mushy zone 20 times larger as compared
to that of liquid steel. As a first approximation, in the present
study as well, the same procedure has been applied and the same
value has been prescribed to the viscosity of liquid in the mushy
zone. However, it is important to assess the sensitivity of
overall predictions to the choice of other possible values of
mushy zone viscosity.
Fi9»3.10 shows the influence of various values of prescribed
mushy zone viscosity on the predicted shell thickness. These show
that over a narrow range (i.e. 20 - 30 times) the overall
influence was only marginal, whereas over a somewhat wider range
(i.e. 20 - 60 times), the mushy zone viscosity affected the shell
growth relatively significantly, particularly, in the lower pool
region. Such behaviour can be attributed to a gradually reduced
bulk flow in the mushy zone with increased viscosity, leading to
a decreased convective transport of heat. Consequently, decreased
shell thickness was predicted with increasing mushy zone
viscosity. Since lower in the pool the mushy zone becomes
gradually thicker, the effect was found to be relatively more
pronounced in the lower part of the liquid pool (Fig. 3. 10).
Shell thickness, mm
143
Distance below meniscus, m
Fig. 3.10: Influence of mushy zone viscosity value on the
predicted shell profile.
144
3.5.4 Influence of Thermal Buoyancy Force on the
Computed Results
In the previous theoretical studies^^' on
fluid flow^ the influence of thermal buoyancy force has not been
taken into account, although buoyancy at the first sight appears
to be one of the principal driving forces for liquid steel flow,
particularly, in the relatively stagnant lower pool region. In
the present study, an attempt has been made to quantify the
influence of buoyancy on the numerical predictions by
incorporating a buoyancy force term (e.g. -“pgiS(T-T)) into the
axial direction momentum equation ( Eqs.(3.2) and (3.3)).
The value of the coefficient of volumetric expansion (/3) ,
embodied in the expression of buoyancy, is not readily available
for liquid steel in the literature^^ ' Therefore, an
estimated^ value of jS = 0.001 “c” was used in all numerical
computations to deduce the buoyancy force originating from
temperature gradients in the liquid pool. To assess the
sensitivity of /3 (the value of which as applied to the present
investigation has some uncertainty) and hence the thermal
buoyancy on the computed results, few calculations with other
possible values of ^ were also carried out. Influence of
different values of coefficient of volumetric expansion on the
estimated shell thickness is presented in Fig. 3. 11. The predicted
shell profiles (Fig. 3.11) revealed that the buoyancy induced
natural convection in the liquid pool has practically negligible
influence on the overall heat transfer and solidification
phenomena in CC.
145
Fig.
shell profile.
146
3.5.5 Role of Prescribed Temperature vs. Insulated
Surface, Out Side the Pouring Stream, as Meniscus
Boundary Conditions
The objective of the present exercise was to
evaluate the appropriateness of the above mentioned boundary
conditions as applied to solve the governing heat flow equations
and consequently , to investigate their resultant influence on the
computed results. As illustrated already, in the first type of
boundary condition, temperature inside the pouring stream was
specified by prescribing the casting temperature (T ) , whereas,
o
ou'tside the pouiring stream, the melt surface was assumed to be
covered with an insulating slag layer and consequently, the
normal gradient of temperature (i.e. the heat flux) was assumed
to be zero. In the other type of boundary condition, casting
temperature was prescribed throughout the entrance boundary (e.g.
on the meniscus, from the line of symmetry to mould wall) . The
influence of these two types of boundary conditions on the
computed results are illustrated in Fig. (3.12) where the
variation of shell thickness with distance below meniscus is
presented. There, it is at once evident that both type of free
surface boundary conditions produce practically identical
results. This clearly suggests that either of the boundary
conditions can be applied to the governing heat flow equation for
estimating temperature fields and the resultant solid shell
profiles in CC billets. These obviously correspond to situations
in which there is minimal or no submergence of the inlet nozzle
below the meniscus.
Rg.
5 12- Influence of two different types of meniscus boundary
conditions (applied to the temperature equation) on th
predicted shell profile.
3.5.6
148
Predicted Flow Field Within the Liquid Pool of
Solidifying Castings
Theoretical studles^l'^« as well as high
temperature experimental study^^'^^ o„ fieya in the liquid
pool of CC have revealed that the liquid pool can be divided into
two principal regions; an upper region in which turbulent
recirculatory flow is essentially induced by the ittomentum of the
incoming pouring stream, and a lower region in which the liquid
is relatively stagnant with natural convection and solidification
shrinkage providing the main driving force for liquid steel flow.
Radio active tracer measurements on a typical billet caster
have appeared to suggest that the flow was predominant (i.e. the
well mixed) only in the upper pool region (i.e. up to a depth of
about 3m ), whereas, significant portion of the liquid pool was
relatively stagnant. Similarly, water model study also appears to
indicate that the depth to which the upper well mixed region
extends below the meniscus depends on the nozzle type, pouring
rate and section size. In billet casting, with straight-bore
nozzles, maximum penetration depth has been reported to be around
4 to 6 times the mould width^^. Furthermore, with straight-bore
nozzles, the flow of liquid steel is downward in the center of
the billet due to the action of the input stream and upward
11 34
(i.e. reverse flow) near the solidification front ' . On the
other hand, the flow pattern in the mould with a submerged
multi-hole nozzle consists of two distinct recirculating loops
for each hole; one stream flowing upward (rotating clockwise)
towards the meniscus while, the other flowing downward (rotating
anti clockwise) . In general, with radial flow multi-hole nozzles.
149
Fig. 3.13; Schematic representation of the flow field with
(a) radial flow nozzle and (b) straight bore nozzle.
150
the upper region ot good mixing becomes much smaller,
consequently, details of flow can be expected to be intricately
related to the nozzle configurations applied to the CC operation,
schematic representation^® of flow fields with radial flow nozzle
and straight -bore nozzle are shown in Fig. 3. 13 .
In the present study, flow fields have been computed for
straight nozzle with no submergence. This in principle
corresponds to open stream casting. Relevant numerical data for
computations are presented in Tables 3.1 and 3.2. Computed
velocity fields at the central vertical plane in round and square
billets casters are presented in Figs. 3.14 through 3.16. There,
for the sake of clarity only a portion of the central vertical
plane is shown. These illustrate that the flow is predominantly
in the axial (downward) direction in the central core of the
billet section, whereas, the flow is directed vertically upward
(reverse flow region) adjacent to the solidification front.
However, very close to the solidification front flow is almost
insignificant. Furthermore, recirculation zone is seen only in
the upper pool region only. It was found to confined only up to 1
to 1.4 m pool depth. Beyond 3 m pool depth flow was almost
insignificant. In the absence of any detailed earlier equivalent
study no extensive comparison can be drawn with the present set
of computed results, although the general nature of the computed
velocity field appears to be consistent with those reported in
literature^^ ,35,36^
151
Inlet Velocity = 1.05 (m/sec)
flow field in a typical
Fig. 3.14: Computed tvo dimensional
round billet” (data set 3, Table 3.1).
152
Fig.
Inlet Velocity = 1.014 (m/sec)
1 = 0
mold exit
Z = t325 ni
3.15: computed flow field in the central
vertical plane of a typical square
billet”(data set 1, Table 3.1).
154
3.5.7 Comparison of Numerical Predictions with Reported
Experimental Measurements
Solidified shell profiles for various casters
(viz., Table 3.1) were derived from the corresponding predicted
temperature fields (see the flow diagram of the computer
program). As shown in Fig. 3. 17, the predicted shell thickness
revealed a discontinuous growth of shell along the casting
direction. This discontinuity in shell growth may be attributed
to the thermal instability prevalent at the solidification front
leading to uneven growth of shell during the casting process. It
is also not unlikely that such a behaviour to some extent may
also be due to relatively coarse grid (spacing 40-50 mm) applied
in the axial (Z) direction in the numerical computations. To
assess these, attempts were made to refine the grid
configurations. This however, had only limited success from the
view point of convergence, because of the reasons enumerated
already. Thus, calculations were carried out with an optimum grid
configuration and relaxation parameters, derived by trial and
error, for each individual casting configuration (Table 3.1)
Fig. 3.17 shows computed shell thickness as function of
33
distance below meniscus for a round billet . Since computed data
points exhibit some scatter, it was decided to smoothen the curve
by regression analysis. The resulting best fit curve was employed
for subsequent discussions.
Figures. 3. 18 through 3.20 present comparison between the
predicted shell thickness and corresponding experimental
measurements reported in literature ' • Reasonably good
agreement all along the pool depth, in all the three types of
Shell thickness, mm
Fig. 3.17: Comparison between the computed shell thickness
and the corresponding best fit curve for a
typical round billet” (data set 3, Table 3.1).
156
Fig. 3.18: Comparison between the present estimates of
the shell thickness end the corresponding
experimentoL meosuremont of a typical round
billet caster (conditions as in ref.11)
Shell thickness, mm
Fig.3.19: Comparison between the P'’®sent estimates of the
^ shell profile and the corresponding experiment^
measurement of a typical square billet caster .
(data set 1. Table 3.1)
Shell thickness, mm
Fig. 3.20: Comparison between the present
^ shell profile and the corresponding
measurement of a typical square billet cast
(dota set 2, Table 3.1).
159
casters considered is readily evident. Furthermore, for the sake
of comparison, the corresponding predictions derived via
artificial effective thermal conductivity based model (viz . ,
Chapter 2) are also incorporated in Figs. 3.18 through 3.20. It
may be noted that this latter model did not agree well with
experimental data. This was already pointed out in Chapter 2,
Sec. 2.5.7. Such comparisons readily demonstrate the superiority
of the conjugate fluid flow heat transfer model over the
artificial effective thermal conductivity model as applied to the
analysis of heat flow phenomena during continuous casting of
steel .
To illustrate the variations in results as shown in Figs.
3.18 through 3.20, predicted temperature profiles at identical
axial locations by the two theoretical modelling approaches
(viz., the artificial effective thermal conductivity model and
the conjugate fluid flow and heat transfer model) are shown in
Fig. 3. 21. It may be noted that the conjugate fluid flow and heat
transfer model predicts nearly uniform temperature field within
the liquid core, whereas, substantial temperature gradients exist
in the mushy zone as well as in the solidified shell. Similarly,
predictions derived via the artificial effective thermal
conductivity model shows similar temperature distribution
particularly in the central pool region, which is the natural
outcome of assuming, a high value of thermal conductivity in the
liquid region.
Nevertheless, a comparison between the two sets of
predictions clearly indicate that energy transport from the
liquid to the solid are considerably different for the two set of
Temperature ,
Fig. 3.21. Comparison between the temperature profiles
predicted by conjugate fluid flow heat transfer
model and effective thermal conductivity model
at the mould exit of a round billet caster .
161
predictions and this in turn appear to suggest that although a
large thermal conductivity assigned to the liquid region leads to
expected thermal gradients in the pool region nonetheless, cannot
simulate the actual energy transport in the entire domain.
Consequently, for investigating thermal phenomena of relevance to
continuous casting, a mathematical model such as the one based on
the concept of artificial effective thermal conductivity appears
to be rather too simplistic and hence, a more sophisticated
approach such as the one considered in the present study (viz.,
conjugate heat-fluid flow model) will be more appropriate.
Finally, in spite of the two dimensional nature of the model
and several approximations involved in developing the model, it
is evident from the predicted results and their subsequent
comparison with the reported industrial measurements, that the
computational procedure developed in the present study can be
conveniently applied to the analysis of various thermal
phenomena in industrial continuous casters.
It may be added here that some macrostructural and
macrosegregation measurements, reported in chapter 4, were
correlated successfully with predictions based on the model. It
has been discussed fully in Sec. 4.5. This is taken as another
confirmation of the reliability of the conjugate fluid flow-heat
transfer model. It has also been proposed that this model may
also be employed to predict equiaxial zone size in a CC billet.
162
3.6 SUMMARY AND CONCLUSIONS
In the present study a steady state, two dimensional (for
the phenomena occurring at the mid face and on the central
vertical plane) mathematical model based on the concept of
conjugate fluid flow and heat transfer has been developed for
continuous casting of steel. Two dimensional turbulent Navier
Stokes equation has been considered for the simulation of fluid
flow in the liquid pool and furthermore, a thermal buoyancy force
term has been incorporated in the axial direction momentum
balance equation to take into account the natural convection
phenomena taking place in the liquid pool of the solidifying
casting. The turbulence properties in the system was estimated
via the Pun - Spalding formula, based on which the average
effective viscosity was computed. Similarly, in the mushy zone,
resistance to the flow produced by the solid matrix has been
taken into account by increasing the viscosity to 20 times the
molecular viscosity of liquid steel. In conjunction with these
considerations, appropriate energy balance equation was
considered, in which the latent heat of solidification was
estimated from the solid fractions in the mushy zone assuming
equilibrium solidification of steel.
The TEACH-T computer code, with considerable modifications,
was used for the numerical solution of the governing fluid flow
and heat transfer equations and thus, to deduce flow and thermal
fields in continuously cast steel billets.
Prior to carrying out any comparison with experimental
measurements, influence of various approximations applied to the
mathematical model were analyzed computationally. Towards this.
163
the predicted shell thickness was found to be almost insensitive
to the precise value of effective viscosity. This in turn
revealed the exact modelling of turbulence in the pool is
relatively less critical than has been originally anticipated.
However, modelling of flow in the mushy zone was found to have
some bearing on the predicted shell thickness, particularly in
the lower pool region. Similarly, influence of buoyancy induced
natural convection on the overall shell growth was found to be
almost insignificant. In contrast, the buoyancy force was found
to have significant influence on the nature of flow field within
the liquid pool.
Velocity and temperature profiles were calculated for three
different CC sections. The predicted velocity fields revealed
that the flow of liquid steel in the pool was predominantly in
the axial direction for most of the central regions, whereas,
near the solidification front some reverse flow were seen.
Furthermore, flow recirculation was found to be significant only
in the upper pool region.
Comparison between predicted shell thickness and
corresponding experimental measurements indicated reasonable
agreement between the two. Similarly, comparison between the
predictions of conjugate fluid flow and heat transfer model and
those derived via the artificial effective thermal conductivity
model demonstrated the superiority of the former over the latter.
The present study has demonstrated that the conjugate fluid flow
and heat transfer approach of modelling is relatively more
accurate in simulating various relevant transport phenomena in
continuous casting in comparison to an ecjuivalent model based on
164
the concept of artificial effective thermal conductivity (viz.,
Chapter 2 ) .
165
Table 3.1; Casting Conditions Considered for Numerical Simulation
Parameters
Data Set l
[ref. 22]
Data Set 2
[ref .22]
Data Set 3
[ref .33]
Cast geometry
square
billet
square
billet
round
billet
Section size (m x m)
0.14
0.133
0.115
Pouring nozzle dia (m)
0.025
0.025
0.02
Mould length (m)
0.51
0.685
0.5
Casting speed (m s~^)
0.0254
0.044
0.0317
Steel carbon (pet.)
0.1
0.1
0.1
Melt superheat* (°C)
25
25
25
Solidus temp. (°C)
1496
1496
1496
Liquidus temp. (°C)
1529
1529
1529
Spray heat transfer
coefficient (W m'^c"^)
650*
650
1079.45**
Caster length simulated
(m) 10
10
10
• - estimated
•• - source ref. [11]
166
Table 3.2: Thermophysical properties of steel* used in the
numerical computations
Density of liquid steel
kg
7200.0
Viscosity of liquid steel
kg s**^
5xl0”^
Coefficient of volumetric^
expansion of liquid steel
H
1
o
o
IXlO"^
Latent heat of solidification
J kg~^
271954
Specific heat
J kg“^c“^
682.0
Thermal conductivity
-1 ~l
W m -^C
34.60
&
15.89+0. OUT
* - source: Ref. [54]
t - source: Estimated data Ref . [55]
CHAPTER 4
STUDY ON MORPHOLOGY AND MACROSEGREGATION IN CONTINUOUSLY
CAST STEEL BILLETS
4.1 INTRODUCTION
Solidif ication of steel in continuous casting takes place
vith columnar and equiaxed dendritic structure. During
solidification, segregation of solute elements (e.g. C,S,P, and
Mn) occurs on both micro and macro scale^®"’^®. Microsegregation
results from freezing of solute enriched liquid in the
interdendntic spaces. But it does not constitute a major quality
problem. Mostly, the effects of microsegregation can be removed
during subsequent soaking and hot working.
Macrosegregation, on the other hand, is nonuniformity of
composition in the cast section on a larger scale. A high degree
of positive segregation in the central region of a continuously
cast section is commonly observed. Figure 4.1 presents one such
typical carbon segregation pattern in a continuously cast slab .
It is established that macrosegregation occurs due to movement of
solid and liquid phases in the mushy zone during final stages of
solidification. The problem of axial or centreline segregation
las been found to be more serious, particularly in high carbon
steels cast at high speed and/or high tundish superheat .
lacrosegregation of solute elements, especially carbon, along the
lentral axis of the cast section results in inconsistent
transfoirmation products (e.g., martensite, bainite) during
168
Ctnfra
u.o
o
o.
1 1 1 1 \ T"
1
^1 1 1 1 1 1
Z
8
§0.6
^ f
-
CD
CC
5
1 L t 1 1 1
20 40 60 80 100 120 :
140 160 160 200 220 240 260
DISTANCE FROM LOWER SIDE, mm
Fig. 4.1; Typical concentration profile as observed
59
in continuously cast slabs .
169
subsequent hot working, and causes nonuniformity in mechanical
properties of the finished product. Also, centreline segregation
is known to be the prime source of sub-surface cracks and
porosity in continuously cast products.
In longitudinal sections, the macrosegregation usually
appears as regular V—shaped lines or bands. Also, the segregation
profile is not smooth but is marked by random oscillations. In
the recent years, there has been a growing concern for another
type of segregation called 'semi-macrosegregation' or 'spot'
61“"63
segregation . The spots are solute enriched regions of sizes
larger than 100 microns and are in between micro and
macrosegregation in size. Semi-macrosegregation spots are known
to be the main source of hydrogen induced cracking in steels
resistant to sour gas.
4.2 LITERATURE REVIEW
Macrosegregation in continuous casting of steels, especially
the centreline segregation, has been reviewed in literature from
time to time^^”^®. Hence, in this review, some of the features
would be dealt with only in brief for the sake of completeness.
Special emphasis would, however, be given on recent trends and
developments as well as some features which are important, but
have not been adequately covered before.
Segregation during solidification of alloys originates from
the difference in solubility of solute elements between solid and
liquid phases. Solubility of a solute in the solid state is lower
as compared to that in the liquid state. As a result, solute
atoms are continuously rejected by the solidifying dendrites
170
leading to constant enrichment of liquid at the solidification
front with progress of solidification. Therefore, the liquid that
solidifies in the final stage may contain significantly higher
solute concentration than its original composition, and on
solidification gives regions of high positive segregation.
Rejection of solute by the solid and gradual enrichment of the
former in the liquid at the solidification front has been termed
as 'zone refining action' 58,64 this phenomenon is
utilized in zone refining of metals.
Redistribution of a solute element between solid and liquid
phases during solidification under equilibrium conditions is
given by the value of its equilibrium partition or distribution
coefficient (k^) , defined as :
where C is the concentration of a solute in the solid in
equilibrium with that in the liquid. The equilibrium partition
coefficient is an important parameter for judging the segregation
tendencies of solute elements in a given alloy system. The
equilibrium partition coefficients are mostly less than 1. Table
4.1 presents some value of k for Fe-binaries.
0
Under the condition of complete mixing in the liquid phase
and no solid state diffusion of solute, the following solute
redistribution equation, which is the well known Scheil's
equation^^'®^'®^, is obtained.
k -1
(4.2)
171
Where is the initial concentration of the solute in the liquid
at the beginning of solidification, is concentration in liquid
during progress of solidification at fraction of solid, f .
s
How6V6r / during real solidification^ complete mixing in the
liquid is hardly achieved. Also, there is always some amount of
solid state diffusion. Moreover, Scheil's equation fails as solid
fraction approaches l (i.e. complete solidification), since C
approaches infinity®^ ' .
Table 4.1: Values of for solidification of iron^'^°
Element
Solid phase
5-Fe y-Fe
A1
0.92
-
C
0.24
0.36
Cr
0.95
0.85
H
0.32
0.45
Mn
0.84
0.95
Mo
0.80
0.60
Ni
0.80
0.95
N
0.28
0.54
0
0.02
0.02
P
0.13
0.06
Si
0.66
0.50
s
0.02
0.014
Ti
0.14
0.07
V
0.90
—
In spite of these limitations, Scheil's equation has been
applied with limited success, in the analysis of micro and
. , 62,66
semi-macrosegregation during continuous cas ing
subsequent studies, Scheil's equation has been modified by Broady
172
and Flemings®’, and also by dyne and Kurtz®®, who took into
account the solid state diffusion in their model. However, the
Clyne-Kurtz equation has been found to be more reliable for
modelling microsegregation in continuously cast slab®®.
The zone refining action, mentioned above, which leads to
continuous enrichment of liquid with progressive solidification
cen also be described by the Burton^s equation®^ ^ ®^ , presented
below. In derivation of this equation, incomplete mixing in the
liquid phase has been considered in contrast to Eq. (4.2).
C
L
(1 - fs>
k -1
eff
. . . (4.3)
in which, k^^^ is effective partition coefficient defined by
69
Burton et al as follows :
where, k is mass transfer coefficient, and takes into account
m
the influence of bulk convection on segregation. R is the linear
growth rate of the solidification front, k^^^ is an important
parameter for describing segregation during real solidification
processes. Eqs.(4.3) and (4.4) predict that the degree of
segregation increases with decreasing growth rate (R) and
increasing k^, which again increases with increasing intensity of
bulk liquid flow. Under conditions of low R or high k^, k^^^
approaches k^, and Eq.(4.3) reduces to Scheil's equation (i.e.
Eq.(4.2)). At the other limit (i.e. high R and/or low k^) , k^^^.
approaches 1, and the steady state situation, where (i.e.
no segregation) is obtained. Equation of Burton et al has been
173
found to be quite successful in modelling axial segregation
resulting from turbulent convection in plane front growth®^.
However, this equation is more appropriate for microsegregation
and semi—macrosegregation in ingots and continuous casting^ It
is not possible to predict centreline macrosegregation from this
equation alone, since macrosegregation is caused by
microsegregation as well as large scale movement of segregated
liquid and solid phases during solidification.
There are several causes leading to movement or transport of
segregated liquid in the mushy zone^^'^^. These are suction due
to solidification shrinkage, change in density of liquid due to
composition change, natural and forced convection in the liquid
pool, turbulent diffusion, movement of liquid and solid phases
due to bulging. Settling of free crystals of steel, besides
causing negative segregation, also induces flow in the bottom
region of liquid pool^^. Flemings et al^^'”^^ carried out
theoretical analysis of macrosegregation in ingots resulting from
flow in inter dendritic channels of the mushy zone due to
solidification shrinkage only. Solid state diffusion and other
causes of flow were ignored. They considered the mushy zone as a
porous medium and applied Darcy's law to evaluate the flow
through complex interdendritic channels, as follows:
...( 4 . 5 )
in which, 7P is pressure gradient in the mushy zone, g is
acceleration due to gravity, is density of liquid, and is
volume fraction of liquid estimated by the following correlation:
174
...(4.6)
X in Eq.(4.5) is the permeability of the mushy zone, which has
been assumed to be a function of and dendrite arm spacing (d,)
as:
^ ... (4.7)
Finally, the investigators proposed the following local
solute redistribution equation for calculation of
macrosegregation resulting from transport of solute-enriched
liquid to feed the solidification shrinkage and thermal
contraction.
ac 1-k^
L 0
+ V
. . . (4.8)
where, VT is temperature gradient, is rate of temperature
change, and /3 is volumetric solidification shrinkage, defined as:
. . . (4.9)
Flemings et al^^ calculated segregation profile of copper in
Al-4.5 pet. Cu alloy ingot, and obtained reasonable agreement
between their theoretical prediction and experimental data. The
investigators, however, considered segregation due to
interdendritic flow only, and influence of bulk flow was
completely ignored.
Under steady state solidification with planar front moving
71 74
with velocity R in X-direction Eg. (4.8) becomes ' :
175
fit = - !:£ fi +
. . . (4.10)
where is fluid velocity in a direction perpendicular to the
solidification front. Eq.(4.10) reduces to the Scheil's equation
when and ^ both are equal to zero. In general, Eq.(4.l0) can
be integrated to the following form^^:
(1-fs)
. . . (4.11)
in which a parameter ^ has been introduced that takes into
account the influence of fluid flow on macrosegregation during
. . . 74
solidification
Equation (4.11) has been termed as 'modified Scheil's
equation' in the present investigation. For ^=1, Eq. (4.11)
becomes identical to the Scheil's equation (Eq.(4.2). For 1,
becomes lower than the value calculated by Eq. (4.2), which
indicates that negative segregation may occur. In the case of 0 <
? < 1 positive macrosegregation occurs. As mentioned already,
complete mixing is practically not encountered during real
solidification process. Therefore, the modified Scheil's equation
is more realistic segregation model than the original Scheil's
equation.
75
In a subsequent study, Ridder et al considered the
interaction between interdendritic flow and fluid flow in the
liquid pool ahead of the liquidus isotherm in their
macrosegregation model. Using experimentally determined
temperature data on Sn-Pb alloys, it was reported that fluid flow
176
in the liquid pool, due to natural convection, had little effect
on interdendritic fluid flow and the resulting macrosegregation.
The investigators recommended their model for study of
macrosegregation in continuous casting and electro-slag
remelting. Besides the above-mentioned models, several other
models have been reported. A good review on this subject is
available®^.
In connection with centreline segregation in continuous
casting of steel, Miyazawa and Schwerdtfeger^® were the pioneers
to model macrosegregation due to bulging. Miyazawa et al^^
obtained a reasonable agreement between theoretical predictions
and experimental measurements. The investigators found bulging to
be the main cause of centreline segregation in continuously cast
slab. It was subsequently confirmed by other investigators. Role
of bulging is well established.
In addition to the theoretical models, semi-empirical models
74 76
of macrosegregation have also been proposed' ' . Takahashi et
al have proposed a semi-empirical model of macrosegregation for
steel ingots. It is based upon the fact that bulk liquid flow
affects morphology and segregation during solidification. Bulk
liquid was assumed to penetrate the mushy zone of columnar region
and sweep out the solute-enriched interdendritic liquid resulting
in a negative segregation. This was called 'washing effect', and
was thought to be the principal mechanism for the formation of
'white band', commonly observed in continuous casting with
electromagnetic stirring (EMS) If the segregation level of
bulk liquid is higher than that in the interdendritic liquid, the
washing effect may lead to higher segregation. On the basis of
177
concept of washing effect, Takahashi et proposed the
following correlation for the effective partition coefficient.
k
eff
1
V
R
. . . (4.12)
in which, B is an experimental constant, is primary dendrite
arm spacing, L is thickness of solidifying zone (mushy zone) , f
wXX
is the maximum solid fraction below which the washing effect
acts, and v is velocity of liquid.
7 6
Takahashi et al also carried out experiments with molten
steel in the laboratory. Freezing was done on a water-cooled pipe
rotating at known and variable RPM. On the basis of the results,
they evaluated the constants and proposed the following
correlation :
k^^^ = 1 - 1.33 X lo"** (1 - kjj) (1 - fgjj) 5 ...(4.13)
Typical value of f^j^ = 0.67, has been reported by the
investigators*^^. As evident from Eq. (4.13), with decreasing
growth rate (R) , k^^^ decreases and consequently extent of
segregation increases. Takahashi et al obtained a reasonable
correlation between their model predictions and experimental
measurements. Eq.(4.13) was semi-empirically derived on the basis
of controlled laboratory experiments. However, its application to
continuous casting has been recommended by the
investigators^^ ' .
178
4.2.1 Influence of Morphology of Cast Structure on
Macrosegregation
In general, segregation is closely related to
.orphology of the oast structure^''^^-’^. parameter that
influences morphology will also influence the macrosegregation
pattern. Structure of plain carbon continuously cast steel
section has three zones, viz., chill zone, columnar zone, and
eguiaxed zone . Figure 4.2 presents a typical macrograph of CC •
billet. Growth of columnar crystals occurs due to constitutional
supercooling associated with the rejection of solute at the
solidification front leading to constant enrichment of residual
liquid. Considerable information is available in
56 64 74
literature ' ' on theory of columnar-to-equiaxed transition.
Studies are mostly with reference to ingot casting, sometimes
unidirectionally solidified. Hence their applicability to
continuous casting is to be always kept in mind. Some findings
are applicable, some may not.
It has been generally accepted in the last two decades that
eguiaxed grains grow on seed crystals already floating in the
nelt^^'^^. Such crystals come either by detachment of crystals
from chill zone or by remelting of dendrite tips and their
consequent detachment in the columnar zone. Therefore, we are
concerned with growth of crystals. Growth of eguiaxed grains (as
free dendrites) prevent further advance of columnar zone. Since
columnar grains are also simultaneously growing, it is the
competition between the growth rates of the two that governs
columnar-to-equiaxed transition^^. To what extent eguiaxed grains
should form prior to transition is being debated. However, some
179
Fig
4.2: Macrostructure of a low carbon steel billet
180
investigators have opinion that it requires reasonable quantity
of equiaxed grains in order to bring in this transition.
Again equiaxed grains may be classified broadly into two
types;
(i) free crystals, randomly oriented: these can move about;
since they are purer and denser than the liquid they tend to
settle downwards. Free crystals also are responsible for bands of
negative segregation zones (V-shaped bands, white bands) .
(ii) equiaxed grains attached to columnar grains: these
would exhibit less random orientation.
The above discussions point out that the columnar— equiaxed
transition may sometime be quite diffused depending upon the
circumstances .
Formation of equiaxed crystals take place over wide region
at a time. This tends to evenly distribute microsegregated
region, and does not allow the zone refining action to aggravate.
Therefore, a very effective method of minimization of axial
segregation is to obtain a large equiaxed zone around the cast
centre. This is a well-established fact, and hence does not
require presentation of too many confirmatory evidences.
An equiaxed structure is preferred over columnar structure
for other advantages such as easier mechanical working,
prevention of internal cracks and centreline porosity. Hence a
Major objective in continuous casting of steel is to obtain as
large an equiaxed zone as possible, and this is facilitated
jjy56,57,60.
(i) low superheat
(ii) medium carbon steel
(iii) electromagnetic stirring, particularly in-mould
(iv) large section size.
181
4.2.2 Influence of Superheat
As stated earlier, the accepted mechanism is that
growth of columnar zone stops when equiaxed zone starts forming.
There always are innumerable tiny crystals (seed crystals)
floating in the melt. When the superheat is dissipated these
start growing thus forming equiaxed zone. Therefore, superheat
should be as low as possible. It is. ^Iso well established. A low
tundish superheat has been found tof enlarge the equiaxed zone and
lower macrosegregation in the central region (Fig. 4.3).
Heat flow calculations, however, . have revealed that during
continuous casting the superheat is extracted almost completely
77
in the mould and upper sprays . Therefore, influence of
superheat on columnar zone lengtti ^as been attributed to the
influence of superheat on the generation and survival of free
crystals in the mould region, which-^in turn, affect the cast
structure many meters below the jjould . Consequently if the
superheat is high most of the seed crystals remelt easily, and
only a few of them survive and become available to bring about
columnar-equiaxed transition in the lower pool region. Again, too
low a casting temperature may regult into nozzle clogging,
difficulties of inclusion f loat-oufe^ and poor surface quality of
the casting. Thus, the casting temperature should be optimum.
4.2.3 Influence of Electromagnetic Stirring
Stirring helps in dissipation of heat of the
liquid pool due to enhanced convectf''{e heat transfer, and helps
in enlarging the equiaxed zon§. This is the basis of
’‘0 to 20 30 40 so 60 70 eo 80
SIZE OF EQUIAXED ZONE, % of total width
Fig. 4.3; Axial segregation index as a
function of equiaxed zone size
183
Slectrcmagnetlc stirring (EMs) i„ continuous casting which has
bsen found to enhance equiaxed zone and out down axial
segregation.
It has been proposed that EMS also causes breaking and
remelting of tips of columnar dendrites. Broken dendrite tips
provide additional seed crystals for equiaxed zone to form (i.e.
crystal multiplication process) . The equiaxed structure formed
due to EMS has been found to be finer than the one caused by low
superheat and/or low speed casting^®. Therefore, with EMS, a
higher superheat can be tolerated in continuous casting without
causing much harm to the internal quality of cast products due to
centreline macrosegregation and centreline porosity.
However, it has been reported that high carbon steels
exhibit some centreline segregation even with low superheat
and/or EMS^®'^*^. For these steel grades, combined in-mould EMS
and EMS during the final stage of solidification (i.e. bottom
60 79
region of pool) has been recommended ' . Also, with EMS, a
narrow band of negative segregation (white band) forms. Formation
of white band has been attributed to the washing effect or sudden
change in growth rate due to EMS^^'^®. Electromagnetic Stirring
is more commonly applied to the slab caster as a measure to
minimize the centreline segregation. For smaller cross-section
(e.g. billets and blooms) it is less common. In the Western World
use of EMS in billet casters®® is limited to only 5 pet. There
are also reports of some adverse effects of EMS, particularly in
slabs. Agglomeration of inclusions and semi-macrosegregation have
been reported to be aggravated in electromagnetically stirred
slabs. Haida et al®® found EMS to disperse the centreline
184
segregation into isolated semi -macrosegregat ion spots over a
wider central region of a slab. Phosphorus segregation ratio as
high as 1.7 in the semi-macrosegregation spots has been
reported^^.
If section size is large then heat flux in the central
region is low. Consequently there is less temperature gradient in
the melt. This induces a large region to attain freezing
temperature and hence larger equiaxed zone. So far as influence
of carbon is concerned, it is well known that high carbon steels
tend to produce a large columnar zone even with electromagnetic
stirring. It is further aggravated by presence of alloying
elements. It has been explained by the fact that with increase of
carbon and also upon addition of some alloying elements (e.g.
Cr) , the freezing range (i.e. temperature difference between
liquidus and solidus at fixed composition) increases. According
to the theory of constitutional supercooling it helps growth of
columnar zone.
From the above point of view a low carbon steel should
exhibit largest equiaxed zone. It seems that is not borne by
fact. The equiaxed zone is largest at medium carbon (0.3-0. 4% C)
and decreases as carbon content is lowered. The explanation for
this is not clear cut. But it is claimed to be caused by
transition from liquid—^ y-Fe to liquid—^ 5-Fe as the composition
moves from medium carbon to low carbon steel.
4.2.4 Role of Peritectic Transformation
It has been observed that steels of chemical
composition close to the peritectic point are prone to cracking
185
and have poor surface quality. This has been attributed to the
j_a, transformation during solidification. Similarly, carbon
content of steel has been found to have significant influence on
the morphology of cast structure.
82
Mori et al studied macrostructures in continuously cast
steel billots. They found columnar zone length to increase in
ascending order when carbon content of steel was changed from 0.3
pet. to 0.1 pet. to 0.6 pet. In other words, the columnar zone
length was lowest at around 0.3 pet. carbon. In a subsequent
78
study, Samarasekera et al , in spite of large scatter in data
(Fig. 4.4(a)), reported minimum columnar zone length for CC
billets containing 0.2 to 0.38 pet. carbon. This is almost
similar to the findings of Mori et al . For blooms, Miyahara et
83
al observed a sudden jump in columnar zone length at 0.42 pet.
C. However, Irving et al found a minimum equiaxed zone width at
0.3 pet. C which is in much contrast to that reported by Mori et
al®^. Again, Kitamura et al®^ have reported almost complete
absence of equiaxed zone below 0.1 pet. C and above 0.45 pet. C
for steel blooms. In spite of these somewhat differing claims by
various investigators, there is a general agreement that equiaxed
zone is maximum at the medium carbon range (0.3 - 0.4 pet.).
In addition to these, in a controlled laboratory experiment
on iron carbon alloys and 8620 type steel ingots, Hurtuk and
Tzavaras^^ measured mould heat flux, columnar zone length, and
dendrite arm spacing. The investigators®^ also measured the
values of (constitutional super cooling parameter) for
6ach case. Their results indicated that, at 0.1 pet. C (lower
limit of peritectic transformation: 5 + L * 'if) > columnar
Columnar zone length , mm
Carbon •/#
Fig 4.4 Influence of carbon contant of steel on columnar zone
^ 7S
length (a) CC billets j Samarosekera etal.
(b) 8620 steel ingots i Hurtuk and Tzovarus®®
187
zone length, value of g/rV 2, mould heat flux were at their
minimum, whereas, dendrite arm spacing was found to be maximum.
Between 0.1 to 0.6 pet. carbon (approximately peritectic
transformation range), columnar zone length,
heat flux were found to increase, and dendrite arm spacing found
to decrease, with increasing carbon content. Close to the upper
limit of peritectic transformation (0.6 pet. c) , columnar zone
length, G/R and mould heat flux attained their maximum values.
Variation of columnar zone length with carbon content in steel,
as reported by Hurtuk and Tzavaras®®, is presented in Fig.
4.4(b). Since these are the data from controlled laboratory
experiments, scatter was much less than in Fig. 4.4(a).
The investigators explained these by the 5— >r transformation
associated with peritectic reaction. Hurtuk and Tzavaras®^
attempted to explain the influence of carbon content on the
columnar zone length in terms of the influence of the former on
8 1
the mold heat flux, as has been reported by Singh and Blazek
who observed a minimum heat flux at 0.1 pet C and attributed this
to the maximum volumetric shrinkage at this carbon level during
solidification. Consequently, in steels having carbon content
close to 0.1 pet., S—^7 transformation occurs at the highest
temperature or sooner after the solidification starts (i.e. close
to meniscus) . Therefore, the influence of solidification
shrinkage on mould heat flux and in-mould solidification is
maximum at 0.1 pet. carbon. As a result, reduced heat flux and
columnar zone length was observed at this carbon level. However,
with increasing carbon content, the ratio of S/7 phases decreases
leading to less shrinkage, increased heat flux and columnar zone
188
length. Also, the influence of constitutional super cooling
becomes more predominant with increasing carbon®®.
on the other hand, Samarasekera et al"^® have attributed the
role of carbon content of steel on morphology of cast structure
during continuous casting, to the influence of peritectic
reaction on generation and survival of free crystals in the
molten pool of metal. According to them, at the lower carbon
levels (below 0.17 pet. C) large volume shrinkage associated with
5 — >7 transformation allow seed crystals to easily get separated
from the mould wall leading to the generation of a large number
of free crystals which are predominantly of 5- phase. These free
crystals survive more easily due to the large melting range of
5-phase as compared to the y-phase. Also, the peritectic reaction
is diffusion controlled (slower process) , and only a few
5-crystals may undergo peritectic transformation and form
y-phase. Whereas, at the higher carbon ranges, crystallites
formed are predominantly of y-phase having lower melting range
and therefore remelt easily. So, at lower carbon range, these
events lead to generation and survival of a large number of free
crystals. As a result, in lower peritectic carbon range, due to
large number of free crystals present, have shorter columnar zone
(larger eguiaxed zone) than steels in other carbon ranges when
78
cast at the same superheat. However, Samarasekera et al found
minimum columnar zone length at 0.2 - 0.38 pet. C rather than at
0.1 pet. carbon, as expected. They attributed this discrepancy to
the presence of Mn (0.46 - 1.39 pet.), which is a y-stabilizer
and could shift the limits of the peritectic transformation to
78
the observed carbon concentrations
189
4.2.5 Fluid Flow, Bulging and Centreline Segregation
As mentioned already, fluid flow in liquid during
solidification plays an important role in the development of cast
structure and macrosegregation. Solidification is accompanied by
shrinkage in volume at the solidification front. This is one of
the causes of fluid flow due to suction. The solidification front
is never smooth and considerable longitudinal fluctuations occur
in solidification front and concentration profile. As a result
there are locations where columnar crystals come up to the centre
and form bridges. This prevents feeding of shrinkage cavity from
the pool and thereby resulting in the formation of mini-ingots
56 57
(Fig. 4.5) ' .It has following three consequences.
(i) The impure liquid from interdendritic region of
columnar zone gets sucked into the axial region increasing axial
segregation .
(ii) Zone refining action becomes more serious due to lack
of feeding of fresh liquid.
(iii) Centreline porosity develops if feeding is incomplete.
In between the support rolls the strand shell may bulge
outward due to the combined influences of solidification
shrinkage, ferro-static pressure of the molten pool, and pressure
(compression) of the support rolls. It has been established that
bulging of the solid shell increases the centreline cavity
leading to enhanced flow of residual segregated interdendritic
liguid, and aggravate mini-ingotism. It, therefore, increases
59
exial porosity and segregation. Miyazawa and Schwerdtfeger
Mathematically analyzed bulging due to roll pressure and found it
190
DENDRITE GROWTH
GROWTH instability
FORMATION OF BRIDGING
FORMATION OF PIPE
ACTUAL MACROSTRUCTURE
Fig. 4.5: Formation of mini-ingot in
continuous casting
192
Bulging, mm
Fig. 4.6: Influence of bulging on centreline
24
segregation
193
64
Fig. 4.7; Interdendritic fluid flow in continuous casting
(a) limiting case, all flow vertical-no
segregation results;
(bj flow resulting in negative segregation at the
cast centre;
(c) flow resulting in positive segregation.
194
improvement in centreline segregation through adjustment of
roll gap taper has been reported in literature^. Adjustment of
roll gap taper reduces the bulging before complete
solidification, and thereby decreases the extent of
macrosegregation. Also, soft reduction (SR) in the cross section
of slab during the final solidification stage has been found to
be guite effective in controlling the fluid flow in the mushy
zone* Another new technique named 'controlled plane reduction
(CPR) ^ / when applied on CC slabs ^ improved centreline
segregation to such a degree as to eliminate macrosegregation.
CPR seems to prevent bulging completely and at the same time
compensate for the solidification shrinkage. Significant
improvement in centreline segregation and semi-macro or spot
segregation in slabs have been reported in literature .
Again, an enlarged equiaxed zone is of help, since equiaxed
crystals do not interfere as much as columnar crystals with
feeding of centreline cavities by the main pool, where the liquid
does not have much segregation. In this connection fine network
structure of equiaxed crystals is desirable. Treatment of molten
metal with calcium or rare earth have been claimed to decrease
extent of centreline segregation supposedly due to more fineness
in equiaxed grain network structure.
4.2.7 Problems of Quantitative Measurement of
Macrosegregation
view of adverse effect of centreline
segregation on product quality in the CC route, the author along
with some others at the Indian Institute of Technology, Kanpur
195
garrisd out some investigations on the same. The various findings
so far have already been reported elsewhere®®'®^, and hence shall
not be presented here as such. Only few remarks would be made.
In contrast to controlled laboratory measurements/
industrial data are characterized by large scatters. In
macrosegregation studies, non-uniformity of macrostructure and
macrosegregation pattern introduce further scatter and
uncertainty. Examples are:
(i) Equiaxed zone size vs. tundish superheat
VS 86
(ii) Columnar zone length vs. carbon content (Fig. 4.4) '
(iii) Fluctuating nature of segregation profile (Fig. 4.1)
A transverse section, as shown in Fig. 4.2, consists of
chill, columnar and equiaxed zones. Sizes of these zones would
vary depending on the section being examined because of
non-uniformity of structure along the longitudinal direction as
revealed by segregation lines and bands, such as u— segregation
band, v-segregation line and v-segregation band (as shown in Fig.
4.8). A major cause of these features is mini-ingot formation as
discussed earlier (Fig. 4.5). In addition fluid flow pattern is
also responsible. It is further illustrated in Fig. 4.9 by
fluctuating nature of segregation profile along the centreline of
a longitudinal section of CC billet.
Techniques of chemical analysis give rise to further
difficulties in some oases. Fig. 4.9 illustrates this point.
Goyal and Ghosh®^ carried out sulphur printing on longitudinal
section of plain carbon steel billets. Sections had been cut
through centreline of the billets so as to reveal the segregation
patterns at and near the billet centreline.
A dark spot on
196
U- segregation band
V- segregation lines
V- segregation bands
Fig. 4.8: Some features of macrosegregation in
longitudinal section of CC products
(schematic) .
Segregation Ratios
rc and r5
197
Location
Rating
Characteristics
Matching
a
2
faint dark spot with
columnar bridge
N
b
3
predominantly white spot
N
c
3
predominantly white spot
with columnar bridge
Y
d
3
predominantly white spot
Y
e
3
predominantly white spot
at closed end of V-line
N
f
2
black spot with pipe
N
g
1
predominantly dark spot
with columnar bridge
Y
h
1
predominantly dark spot
N
i
1
predominantly dark spot
with columnar bridge
Y
r^r-n-Files of carbon and sulphur along
Fig. 4.9: Segregation profiles
the centreline of a typical steel billet
198
sulphur print indicates region of high sulphur content and white
spot low sulphur content. Then samples for analysis were
collected by drilling with 8 mm dia. drills up to 8 mm depth at
different locations on the axes of billets. Drillings were
subsequently analyzed for carbon and sulphur by automatic carbon
and sulphur determinator. A difficulty of chemical analysis is
that analysis results would depend on choice of drill size^®.
A rating system was developed®^ for correlating chemical
analysis with sulphur print, in this, i corresponds to the
location predominantly dark, 3 predominantly white, and 2 in
between 1 and 3. Matching is denoted by yes (Y) whenever 1
corresponded to peak, 3 corresponded with trough and 2 with
middle position. No matching is denoted by N. It was found that
both match and mismatch were there. One of the causes of mismatch
lay in the fact that drilling had to be done to a finite depth to
collect some sample. Due to non-uniformity it is very likely that
the same drilling location has layers of alternate positively and
negatively segregated regions. Non-destructive chemical analysis
such as X-ray fluorescence or emission spectroscopy can remove
this source of uncertainty. But to the best of author's knowledge
the latter techniques cannot be employed for such large samples.
Previous studies^® revealed that maximum segregation
ratio for sulphur (i.e. ratio of centreline sulphur percent to
estimated average sulphur percent in billet sample) in billet
samples of Indian Steel Plants ranged between 1.2 to 1.8.
Besides some variation in steel composition, the only other
variable that was recorded was tundish superheat. Attempt was
®ade to correlate maximum segregation ratio with superheat.
199
scatter in data did not allow revealing the expected trend. Study
89
by Goyal and Ghosh also indicated the value going up to
approximately 2 or somewhat above for both carbon and sulphur.
4.2.8 Macrosegregation and Mew Measurement Techniques
As stated already in Section 4.1, semi-
macrosegregation (or spot segregation) is drawing World-wide
attention in recent years, and it is considered as a more serious
defect in high grade continuously cast steels. Spot segregation
is reported to form due to transport of segregated liquid in the
mushy zone during the final stage of solidification®^'®^. The
principal cause of transport has been found to. be bulging of the
62 63
strand between the support rolls ' . The interesting feature of
semi-macrosegregation is that it forms even with equiaxed
solidification in the central region. As mentioned already, spot
segregation has been found to be intensified by the application
of electromagnetic stirring and soft reduction, even though the
usual centreline macrosegregation is suppressed with these
measures . In steels for conventional application
semi-macrosegregation has not been considered as a serious
problem. However, in high grade steels for off-shore
applications, even low level of segregation has been considered
to be harmful. As a result of this, the permitted level of
segregation in these steel grades has become more stringent.
This calls for the precise evaluation of segregation in
continuously cast steel. Because of small size of segregation
spots, semi -macrosegregation is difficult to evaluate by the
conventional techniques such as: sulphur print, warm acid etching
200
and chemical analysis, sulphur print technique also fails to
detect macrcsegregation in low sulphur and calcium treated
90
steels . Macroetching with hydrochloric acid has been found to
be inadequate because of its poor resolving power®®. The
difficulties associated with the traditional method of sampling
by drilling and chemical analysis, have been already discussed in
the earlier section . Too much sample is required for
analysis.
In view of the above mentioned reasons, new evaluation
techniques have come up. These include different types of
electron probe micro analyzer (EPMA) , etch print (EP) technique,
. 90
and image analyzer . Traditionally, EPMA has been used only for
evaluation of dendritic microsegregation because of its inability
to analyze large segregation spots. Nippon steel laboratory has
developed a macroanalyser (MA) which is based on the same
analytical principle as EPMA, but it can measure segregation on a
relatively large area (i.e. 20 urn - 5 mm) . Large samples (size
300 X 100 mm) can be accommodated in the instrument, and
quantitative segregation maps of c, Mn and P can be obtained
across the sample section. MA can also provide quantitative
information on the fraction of segregated area and the size
distribution of the segregation spots.
For rapid detection of segregation, etch-print technique has
been developed^ It has been reported that etch print can reveal
even extremely fine details (e.g. fine cracks, segregation spots)
of cast structure. In some studies^^, etch print technique and
image analyzer have both been employed. The combination of etch
print and image analyzer facilitates measurement of total number
201
of spots, size distribution, and area fraction of
semi-macrosegregation spots .
4 . 3 EXPERIMENTAL PROCEDURE
The present investigation on macrosegregation in transverse
section of continuously cast billets involved the following
program:
(i) collection of samples from continuously cast billets,
and corresponding shop floor data
(ii) macrostructural examination of transverse sections of
billet samples to evaluate various morphological
features of billet casting
and (iii) determination of composition of steel at the location
of columnar-to~equiaxed transition boundary and at the
centre in transverse sections, in order to evaluate
macrosegregation and other characteristics of
solidification.
4.3.1 Plant Data and Sample Collection
An important part of the present study was
collection of billet samples and the corresponding data from the
continuous casting shop of steel plant. On the basis of various
aspects of solidification and macrosegregation phenomena during
continuous casting, described earlier, the following factors have
been taken into consideration during sample collection.
(i) As mentioned already, industrial data are
characterized by large scatter, and the continuous
202
(ii)
(iii)
(iv)
(V)
casting process is no exception to these. The way to
obtain a more conclusive pattern is to carry out
investigations on a large number of samples.
In Indian practice, there is no electromagnetic
stirring. Hence, the size of equiaxed zone and
centreline segregation are primarily controlled by
the superheat of the liquid steel. Therefore, in
order to establish correlation with superheat and
morphology and/or macrosegregation, casting
temperature should be measured as precisely as
possible corresponding to each billet sample
collected from the strand.
lasting of a particular heat in general continues for
a long period, and there may be significant drop in
temperature of liquid metal during continuous
casting. Therefore, measurement of liquid steel
temperature should be carried out at least twice, one
at the beginning and another towards the end, and
only the corresponding billet samples should be
collected to study the influence of superheat.
For a multistrand caster, each strand may have
somewhat different characteristics. Therefore, all
samples should be collected from one strand only, m
order to Keep the strand characteristics fixed and to
make the comparative study more meaningful.
d 1-0 carry out investigation with
It is desirable to carry
,, sm of Steel particularly those grades
different grades of steei, p
uiatn of macrosegregation is
in which the problem
203
relatively more serious (e.g. high carbon steels) .
However, as described earlifer, one of the objectives
of the present study has been to correlate the actual
columnar'~eguiaxed transition with the theoretical
prediction of temperature field on the basis of heat
transfer model developed in the present study (Ch.3).
Also, another objective has been to study the
applicability of various segregation models reported
in literature, to macrosegregation in continuously
cast billets. Such studies can be carried out even on
a single grade of steel. Therefore, it was decided to
collect all samples of same grade of steel.
Keeping the above points in mind, 21 low carbon steel billet
samples of approximately 100 mm thick have been collected from
the continuous casting shop of Tata Steel, Jamshedpur. The
specification of the billet caster at Tata Steel is presented in
Table 4.2. More details of the process adopted at Tata Steel are
available elsewhere^^. Additional comments on plant data
collection are presented below.
(i) All samples were collected from the strand number 4
which is one of the central strands.
(ii) For each caster, two billet samples were collected,
one at the beginning and another towards the end of
casting.
(iii) For the precise determination of superheat it is very
desirable to measure the temperature of liquid steel
in the mold or in the tundish-to-mold pouring stream.
However, in industries the temperature of the liquid
204
steel is Measured either in the ladle or in the
tundish.
Hence, as a part of data collection program, special efforts were
made to measure temperature in the mold or in the pouring stream.
For this a Pt~Pt/10pct.Rh thermocouple assembly was designed and
fabricated, and taken to the plant. However, it was not
successful due to fluctuations in the output meter (i.e.
millivoltmeter) and lack of sufficient life of thermocouple in
the liquid metal. Hence, temperature was measured by the routine
immersion thermocouple in the tundish only.
Another effort was made on temperature measurement by
optical pyrometer. However, non-availability of a sophisticated
two-color optical pyrometer did not allow satisfactory
measurements. Hence, it was decided to estimate the casting
temperature from the tundish temperature as closely as possible
by some theoretical estimation procedure as the other alternative
approach.
(iv) Subsequent correlation with superheat required that
the samples be collected from a portion of billet which
corresponded approximately to liquid steel whose
temperature was measured.
This required knowledge of casting speed. In order to get it
precisely, casting speed was independently calibrated several
times . Table 4.3 presents the values of casting speed along with
other data. These values are averages of 2-3 measurements in each
heat. The casting speed, however, varied within a narrow range in
all the heats.
205
Table 4.2: Characteristics of continuous casting machine at TATA
91
STEEL
Machine:
supplier
Type
Radius
Number of Strands
Distance Between Successive
Mould:
Material
Length
Sizes
Lubrication
Oscillation Frequency
Water Flow Rate
Secondary Cooling Zone:
Total Length
Number of Zones
Water Flow Rate
Casting Speed:
Metallurgical Length:
Tundish:
Capacity
Nozzle Diameter
Refractory
Ladle:
Capacity
Nozzle Diameter
Teeming Mode
Sequence Casting:
CONCAST AG, w. Germany
Curved Mould Billet caster
6 m
6
Strand 1100 mm
Water Cooled Chrome
Plated Copper
0.8 m
100x100 mm sq. and
125x125 mm sq.
Rape Seed Oil
100 to 150 Cycles Per min.
504 m^h“^
7.45 m (Approx.)
4
324 m\"^
3 m min. ^ for 100 mm sq.
2.2m minr^ for 125 mm sq.
Billets
16.2 - 19.9 m
12 tonnes
12 mm for 100 mm sq. and
15 mm for 125 mm sq. Billets
Garnex Board; Nozzle-Zirconia
130 tonnes
45 mm
Slide Gate Valve with no
Turret Facility
3 to 5 Heats
206
4 . 3.2 Macroetching of Transverse Section of Billets
The main objectives of macroetching in the present
study have been the determination of area fractions of chill/
columnar, and equiaxed zone, as well as determination of the
position of columnar-to-equiaxed transition (GET) boundary in the
transverse sections of the billet samples. For macroetching
standard procedure described in literature, was adopted.
The billet samples were sectioned to the required size (50
mm thick samples) , and subjected to the surface grinding before
macroetching. Precautions were taken to avoid deep scratches and
machining marks. After machining, sample surfaces were cleaned
with acetone to remove dirt, oil and grease. Subsequent to this,
samples were macroetched with warm 1:1 hydrochloric acid-water
solution (by volume) to which about 10 ml hydrogen peroxide was
added. During macroetching temperature of the etchant was
maintained at 60-65°C, and the duration of etching was kept at 25
min. Also, in order to avoid the initial drop in the etchant
temperature while dipping the sample into the etching solution,
samples were preheated to the etching temperature (i.e. 60-65 C)
in an oven. After etching, samples were washed with ammonium
hydroxide solution in order to remove the acid completely from
the macroetched surface, and final washing was carried out under
tap water. Samples were then dried thoroughly in hot air
and finally, kept in the oven maintained at 80 C in order
protect the macroetched surface from rusting.
After macroetching, macrostructure of each billet sample was
examined under magnascope (magnification
the macrostructure were examined. All
3X) . Various features of
visible macr ©structural
207
features of transverse sections of the billets were traced on
transparent papers. These tracings were subsequently used for
measurements of fractions of chill, columnar, and equiaxed zone
as well as to ascertain the positions of CET boundaries using
transparent graph papers. For checking the reproducibility of the
macroetching procedure each billet sample was macroetched at
least twice and the above mentioned procedure was followed for
the measurements. Finally, macro photographs were taken for each
section.
4.3.3 Chemical Analyses of Samples
Subsequent to the macrostructural examination,
carbon and sulphur contents of steel at the CET boundary as well
as at the centre of each billet section were determined. There
are several methods for determination of carbon and sulphur in
steel. In recent years, new evaluation techniques of segregation
based on electron probe analyzer have also come up. However, due
to non-availability of such sophisticated instruments, more
frequently applied 'drilling technique' was adopted in the
present study. It has been reported that the drilling technique
may underestimate the actual segregation level due to some
averaging effect in chemical analysis, if the drilled volume is
large .
Therefore, a proper sampling/drilling scheme is crucial for
a meaningful evaluation of segregation. In previous studies drill
diameter varied from 3 to 8 mm and the drill depth were kept
between 3 to 14 mm. Considering these factors samples were
drilled out with a 3 mm diameter drill up to 5 mm drill depth at
208
the CET boundary in each billet sample, m order to generate
sufficient quantity of sample, drilling was conducted at 6-8
locations along the CET boundaries, and then these were mixed. At
the centre, most of the billets had centreline porosity. To
generate sufficient samples, drillings were carried out with a
larger (5 mm) diameter drill. Fig. 4.10 presents a sample
photograph of a drilled surface.
Prior to drilling, the billet surface was macroetched and
various locations of sampling in the macrostructure were marked
with a punch. The surface was then cleaned. After drilling,
samples were collected on plastic sheets. Finally, drillings were
washed with acetone and distilled water, dried in oven, and
stored in plastic envelop with proper identification marks.
Chemical analyses of the samples were carried out in the
carbon-sulphur determinator at the National Metallurgical
Laboratory (NML) , Jamshedpur. The instrument was a CS-444
microprocessor-based determinator supplied by LECO, USA. The
instrument is capable of doing measurements of carbon and sulphur
contents of metals, ores, ceramics, and other materials. It is
fitted with CS-444 determinator, the HF-400 induction furnace, a
built-in balance, display monitor, printer Fig. 4.10
and key board.
Analysis of carbon and sulphur required l g sample weight
and the duration of analysis was 1 min. During analysis of
samples, the instrument was calibrated at different stages with
LECO standard sample. For some samples, duplicate analyses were
carried out in order to check the reproducibility of analysis.
209
Fig. 4.10: Photograph of a drill surface.
210
4.4 RESULTS AND DISCUSSIONS
Table 4.3 presents the details of data collected from Tata
Steel for the billet samples. The data consist of temperature of
liquid steel in tundish, casting speed, chemical analysis of
liquid steel (pet. C, Si, Mn, S, p etc), the grade of steel, and
the time interval between temperature measurements in the
tundish .
As stated in section 4.3, the casting speed was determined
by the author for the present investigation in order to make it
as precise as possible. Two billet samples were collected for
each cast, one towards the beginning of casting and another after
the time intervals indicated in Table 4.3. Temperature of molten
steel in the tundish was measured by immersion thermocouple. The
billet samples were so collected as to correspond approximately
to liquid steel for which temperature were measured.
The overall tundish temperature variation in different heats
were from 1525 to 1570 °C as noted in Table 4.3. All the grades
of steel were of low carbon steel with carbon content varying
from 0.08 to 0.2 pet. It has already been stated in section 4.3
that all samples were collected from strand 4 of the caster.
4,4.1 Results and Discussions on Macrostructural
Examination
As stated in section 4.3, the macrostructure was
examined on transverse section of billet only. The examinations
were made under unetched and etched conditions both.
Table 4,4 presents macrostructural details of transverse
211
section of billet samples, m unetched condition 2 features were
noted viz. rhomboidity and existence of pores especially near
billet axis.
4. 4. 1.1 Measurement of equiaxed zone size
In the past the investigators employed either
columnar zone width or equiaxed zone width or area of equiaxed
zone for further correlation with superheat etc.^®'"^®'®®. m the
present investigation it was decided to employ area of equiaxed
zone for further correlation purposes. Table 4.4 presents data.
It may be noted that areas of zones have been presented as pet.
of cross-sectional area of billet.
As stated in section 4.2, each billet sample was polished
and macroetched two to three times, and separate measurements
were carried out after each macroetching in order to find out
reproducibility of the entire procedure, and obtain more precise
values. The reproducibility of measurements was ±5-20 pet. of
average. This variation is attributed to the finite width of
columnar-equiaxed transition zone. The transition is mostly not
sharp and it was not possible to mark the transition boundary
precisely. As discussed in Section 4.2, the colvunnar-equiaxed
transition is expected to be diffused in nature. Hence, this
observation is in agreement with what is expected and has been
observed by others^^'^^.
212
Table 4.3 Data on billet samples collected from Tata Steel
Sample
code
Tundish
temp.
Casting
speed
(m/min)
Nominal composition (pet.)
Time
interval
bet. temp
Grade
of
Steel
( C)
C
Si
Mn
s p
measure-
ments
Al
A2
1540
1535
1.7
1.8
0.20
0.244
0.76
0.03 0.029
50 min.
TMT50
B1
B2
1530
1520
1.7
1.75
0.18
0.229
0.71
0.025 0.029
46 min.
TMT50
Cl
C2
1543
1535
1.8
1.75
0.08
0.129
0.48
0.026 0.02
10 min.
C1008
expo.
D1
D2
1557
1545
1.8
1.7
0.11
0.158
0.69
0.034 0.026
1 hr.
10/13
Si-Ki
El
E2
1550
1538
2.2
2.1
0.20
0.204
0.74
0.033 0.032
1 hr.
5 min.
TC2
FI
F2
1548
1543
1.95
2.1
0.13
0.219
0.68
0.032 0.017
1 hr.
10/13
Si-Ki
G1
G2
1543
1533
1.75
2.0
0.13
0.185
0.69
0.033 0.023-
40 min.
10/13
Si-Ki
HI
1568
1.75
0.09
0.0106
0.50
0.029 0.024
C1008
H2
1555
1.95
Cr=
=0.01, Ni=0.016, Mo=0.01
48 min.
expo.
I
1563
1.75
0.18
0.20
0.73
0.03 0.027
-
J1
J2
1545
1528
1.6
1.65
0.14
0.25
0.70
0.025 0.030
1 hr.
10 min.
—
K1
K2
1550
1535
1.7
1.6
0.11
0.125
0.43
0.033 0.011
1 hr.
10 min.
213
Table 4.4 has a ooluam indicating whether the e,miawed zone
was synmetrio or asyametrlc around the geometric axis of the
Ullet. From Table 4.4 it may be noted that equlaxed zone was
asymmetric around both perpendicular directions parallel to edges
(l.e. r and y directions) for 10 samples. There were 6 samples
Where it was symmetric around the centre, in rest 5 samples, it
was symmetric with respect to only one axis. Figs. 4.11(a) -
4 . 11 (c) show macrograph of billet sections. Figs. 4.12(a) -
4.12(c) present the sketches of corresponding CET boundaries. It
may be noted that equiaxed zone is symmetric in Fig. 4.11(a),
whereas, it is asymmetric in Figs. 4.11(b) and 4.11(c).
78
Samarasekera et al have also reported asymmetric equiaxed
zone in curved mold CC machine. The investigators observed longer
columnar structure adjacent to the inside radius face than that
next to the outside radius face. They attributed this to the
preferential settling of free crystallites due to gravity at the
columnar solidification front advancing from the outside radius
face. These free crystallites subsequently, interfere with the
columnar growth, and lead to asymmetric equiaxed structure in the
solidified billet. The CC machine at Tata Steel is also a curved
mold type. The observed asymmetry in structure here may also be
partly due to the above explanation provided in literature. But
it is not possible to make further statements about the exact
cause of asymmetry here.
214
Tabl® 4.4; Measured Morphological Features in Transverse Section
of CC Billets
Features of unetched
surface
Area percent of various zones on
etched surface
El 0.2
E2 0.2
.13 28
.13 21
.13 24
.13 13
.09 42
.09 27
.09 42 3 mm dia
.09 27 small
II 1 0.18 48 2 mm dia
m
27
K2 10.111 12
1.02
1.03
1.05
Chill
Equiaxed
11.0
14.0
9.0
10.0
10.0
8.0 9.0
13.6 12.5
13.0 12.0
12.0 15.0
11.4 13.0
12.5 9.5
13.0 8.0
10.5 12.0
11.0 11.5
10.0 8.0
9.0 10.0
8.0 10.0
10.0
10.0 11.0
20.0
28.5
28.0
31.0
18.0 23.0
24.5 31.0
24.0 32.0
36.0 33.0
2.5
8.0 9.0
13.0 24.8
12.5 36.0
13.5 30.0
12.0 42.0
3.5
.5| 9.5
6.5
15.5
5.5| 4.0
8.0
21.5
32.0
26.5
42.0
6.5
10.5
8.6
12.8
9.0 9.0 13.0
215
(a)
]/■ .Vr -:
Fig. 4. 11: Photographs of macroetched surface of billet
samples with equiaxed zones as follows:
(a) symmetric (type I)
(b) asymmetric about one axis (type II)
(c) asymmetric about both axes (type III)
217
4 . 4 . 1.2 Influence of tundish superheat on equiaxed zone
size
It has already been discussed in literature review
(Sec. 4.2.2) that more the superheat, smaller would be the
equiaxed zone size. It has been verified by many investigators.
Fig. 4.13 shows the area of equiaxed zone in percent of total
cross-sectional area as a function of tundish superheat (AT, in
°C) . AT is defined as:
V \ ...(4.14)
Where is temperature of liquid steel in °C as measured in the
tundish. is liquidus temperature of steel in °C at its nominal
composition (composition of liquid steel as collected from
plant) .
Steel is a multi-component alloy. Therefore, its liquidus
temperature cannot be estimated precisely from the binary
iron-carbon phase diagram. The standard approach of estimation of
liquidus temperature of multi component alloys have been to sum
the depressions which each component element would impose on the
melting point of pure iron according to the respective binary
phase diagrams. However, this approach is valid for dilute
solutions, with negligible interactions amongst the solute
elements, or 'quasibinary^ alloys where only one of the elements
. 92
is non-dilute and provided the solid phase is the same .
In literature^^”®®/ various correlations ’ between the
liquidus temperature and composition of steel have been proposed.
QO 93
A good review on this subject is available . Thomas et al have
reported one of the correlations for liquidus temperature, which
is noted below:
218
1537 - 88(pot.C) - 25(pct.S) - 30(pot.P) - 8(pct.Sl)
-5(pct.Mn) - 5(pct.cu) - 2(pet.Ho, - 4(pct.Ni)
-1.5(pct.Cr) -18(pct.Ti) - 2(pct.V)
• • ■ (4.15)
in the present study, the liquidus tenperature of steel has
been estimated using various correlations reported in literature.
Table 4.5 presents the results of calculations. It is to be noted
from Table 4.5 that the values of predicted from Eg. (4.15)
.atoh closely with those of Howe’^. But there is some mismatch
wibh othsirSe Therefoire/ corirelation of Thomas et al^^ (Eg. 4, 15 )
was selected for the subsequent calculations of the liquidus
temperatures .
The calculated values of tundish superheat (AT) for various
samples have been presented in Table 4.4. Fig. 4. 13 shows the
variation of area pet. equiaxed zone with AT, and confirms the
established literature finding that area of equiaxed zone
decreases as the tundish superheat increases. Roy et al^® earlier
reported this on some Tata Steel billets. However their data were
limited. In Fig. 4. 13 data collected by Roy et al^® have also been
included for the sake of completeness.
As may be noted from Fig. 4. 13 that there is lot of scatter
in data points. The issue of scatter has already been discussed
in Section 4.2, and it has been shown that such scatter is a
characteristic feature of these industrial data as reported by
others in literature. In order to establish the trend, linear
tfigression analysis was done, and the best fit line is shown in
Table 4.5
Estimated llquidus temperatures of billet
samples using different correlations
219
\ 1512.0 1515.2 1510.4 1515.1 1511.4
B 1514.3 1517.5 1512.7 1516.7 1513.8
C 1525.3 1527.7 1524.9 1524.3 1524.8
D 1521.0 1523.8 1520.3 1520.9 1520.3
E 1512.3 1515.6 1511.0 1515.5 1511.7
F 1519.1 1521.9 1517.7 1519.5 1518.5
G 1519.1 1522.1 1518.1 1519.7 1518.5
H 1524.9 1528.2 1525.4 1525.2 1524.6
I 1514.3 1517.7 1513.0 1516.9 1513.8
j 1517.7 1520.5 1516.2 1518.3 1517.0
K
1523.0
1526.0
1522.4
1523.2 1522.5
221
Fig. 4 . 13. The equation of best fit line is:
'^Eq “ 21.45 - 0.17 AT _
where, = area of equiaxed zone as pet. of total cross
sectional area.
4.4.2 Results and Discussions on Macrosegregation Studies
4. 4.2.1 Results
As stated in Section 4.3, the chemical analysis of*
liquid steel for each cast was provided by Tata Steel. The plant
takes lollipop samples and analyze them in their Express
Laboratory of steelmaking division by spectroscopic method. These
have been designated as nominal composition and reported in
Table 4.3.
For study of macrosegregation, drillings were collected from
centreline as well as from columnar-equiaxed transition (CET)
boundary. The samples were analyzed by LECO carbon-sulphur
determinator at the National Metallurgical Laboratory with
participation of the author. Table 4.6 presents the results of
chemical analysis of all samples. Each analysis represents the
average of a set of duplicate analyses.
The average reproducibility of carbon analysis as determined
from the duplicate sets was ± 2 pet. of the value. For sulphur it
was ± 6 pet. of the average value. The reliability of analysis
was checked frequently by using standard samples. Moreover, few
lollipop samples collected from the plant were also analyzed at
the National Metallurgical Laboratory. They differed by few
percent only of the value from those provided by the plant
16 )
222
Table 4.6: Analyses of carbon and sulphur at the centreline
and CET boundaries of billet samples
Sample
nominal
0.2
0.2
0.18
0.18
0.08
I 0.08
0.11
0.11
0.2
0.2
0.13
0.13
0.13
0.13
0.09
0.09
0.18
0.14
0.14
0.11
0.11
Carbon
centre
line
0 .
232
0 .
302
0 .
18
0 .
225
0 .
096
0 .
108
0 .
102
0 .
122
0 .
204
0 .
237
0 .
,155
0 .
,134
0 .
.171
0 .
.159
0 ,
.105
0 .
.085
0
.192
0
.19
0
.183
0
.231
0
.212
CET
boundary
0.218
0.21
0.197
0.183
0.105
0.089
0.11
0.115
0.233
0.227
0.145
0.139
0.133
0.126
0.087
0.093
0.2
0.16
0.166
0.183
0.188
nominal
0.03
0.03
0.025
0.025
0.026
0.026
i 0.034
0.034
0.033
0.033
0.032
0.032
0.033
0.033
0.029
0.029
0.03
0.025
0.025
0.033
0.033
Sulphur
centre I CET
line boundary
0.0325 0.0203
0.0616 0.0323
0.0344 0.0326
0.0361 0.0261
0.0227 0.0265
0.0266 0.0266
0.0275 0.0323
0.032 0.0354
0.0336 0.0282
0.0325 0.0392
0.042
0.042
0.027
0.027
0.0378 0.0298
0.0381 0.0333
0.0255 0.027
0.0275 0.0293
0.0285 0.024
0.0337 0.0315
0.031 0.0295
0.0374 0.0297
0.0374 0.0318
223
laboratory .
Extent of segregation is typically expressed by a parameter
called 'degree of segregation', which is a measure of level of
segregation defined as:
^i “ ...(4.17)
where Cj^ == concentration of solute element i at the location
under consideration
^io ~ concentration of i in liquid steel (nominal
concentration)
and = degree of segregation of i.
Table 4.7 presents values of degree of segregation of carbon
as well as sulphur (r^ and r^ respectively) for all samples from
chemical analysis data.
4. 4. 2. 2 Comparison of segregation levels at centreline and
at columnar-equiaxed transition (CET) botmdary
Fig. 4. 14 shows the data points of degree of
segregation for sulphur (r ) at the centreline vs. r^ at
columnar-equiaxed transition (CET) boundary. As usual like other
data, there is scatter. However most of the data points lie above
the line with slope 1:1. This demonstrates that statistically
speaking, level of segregation at the centreline was more than
that at the CET boundary. As Fig. 4. 15 shows this was the feature
of r^ as well. This observation is in agreement with what is
expected from theoretical considerations i.e. segregation level
at centreline should be more than that at any intermediate
location on the billet section.
224
Table 4.7; Degree of segregation of carbon and sulphur at the
centreline and CET boundaries of different billet
samples
Sample
No.
Centreli
ne
* 1 ".* —
rr CET
Boundary 1
mmim
^s
Bsgiagsi
A1
1.16
1.21
1.28
0.8
1.13
m
A2
1.51
2.28
2.0
0.72
1.08
2.37
B1
1.03
1.53
13.03
0.72
1.13
1.45
3.04
B2
1.3
1.6
1.79
0.67
1.05
1.16
3.04
Cl
1.23
0.9
-0.51
0.97
1.35
1.02
0.06
C2
1.14
1.04
0.3
0.94
1.14
1.0
0
D1
0.96
0.94
1.52
0.96
1.04
1.06
1.48
D2
1.15
1.04
0.28
0.91
1.04
1.16
3.78
El
1.05
1.28
5.1
0.77
1.2
0.95
-0.28
E2
1.22
1.17
0.79
0.66
1.17
1.33
1.82
FI
1.23
1.47
1.86
0.72
1.15
0.94
-0.44
F2
1.06
1.07
1.16
0.58
1.11
1.22
1.94
G1
1.35
1.27
0.8
0.95
1.01
1,00
0
G2
1,26
1.21
0.82
0.9
1.02
1.12
5.72
HI
1.2
0.98
-0.11
0.93
1.0
1.06
-
H2
0.96
1.07
-1.48
0.86
1.06
1.12
1.94
I
1.1
1.07
0.68
0.89
1.14
1.073
0.54
J1
1.4
1,5
1.21
0.94
1.16
1.4
2.24
J2
1.34
1.48
1.35
0.88
1.22
1.31
1.35
K1
2.16
1.26
0.3
0.81
1.73
1.17
0.3
K2
1.98
1.26
0.34
0.8
1.76
1.07
0.12
fs, CET boundary
Fig. 4.14: Relationship between rs at the centreline and
columnar—equiaxed transition (CET) boundary.
fc, CET boundary
Fiq 4.15; Relationship between rc at the centreline and
CET boundory.
227
4. 4. 2. 3 Quantitative relationship between r and r
s c
As dxsciissGci in ssction A o
!»ecrion 4.2, macrosegregation
depends on (a) equilibrium partition coefficient (k^^) of solute
elements and the parameter, R/k^, (b) morphology, (c) movement
of solid and liquid phases during solidification, and (d) extent
of chemical reactions during freezing, i.e. formation of
inclusions etc.. For a particular location, factors (b) and (c)
are common for all solutes, but not (a) and (d) . Hence, r^ values
of various solutes at a given location would not be the same, but
due to the common factors some correlations amongst them can be
96
expected. Iwata et al plotted the r^ values of s, Mn and P
against that of carbon at the centreline by a linear plot. In
spite of considerable scatter in data, investigators have
reported some correlations of r , r and r with r . Moore^ has
S Mn P C
described the advantages of this particular approach for
determination of segregation ratios of C,S,P and Mn by knowing
the segregation level of one of them. However, sulphur tends to
form inclusions with Mn and other elements. Also, r does not
Mn
show sufficient variation with r^ to provide an accurate
assessment. Therefore, Moore^® has recommended phosphorus to be
the best element to use for determining the degree of segregation
of carbon by using the above mentioned approach.
In the present study, also, it was decided to find out a
quantitative relationship in a similar fashion mentioned above.
Pig. 4. 16 presents data points on r^ vs. r^ plot for the
centreline of all the billet samples along with the best fit
line. The best fit line of Iwata et al^^ as well as that of Goyal
and Ghosh®® are also shown for comparison purposes. The equations
Degree of sulphur segregation (rs)
(wato et ol.*® *■/■
Fig.
4 . 16 : Relotionship between rs and re at the centreline
in billet samples.
Degree of sulphur segregotion (rs)
C£T boundory-
“Centreline
• Exp. data
Best fit lines
3,9 ry ^ r 't r '"T " "'r r- > t m m n i i i i M i »'TTi
0.9 1.1 1-3
Degree of carbon segregation ^.rcj
Fig. 4.17: Relotionship between rc and rs at the GET
boundory In billet samples.
231
different from that for centreline. However they are almost
parallel*
AS discussed in the newt sub-section that no segregation
equation predicts linear variation of r^ with r . It is a purely
empirical approach. Hence no effort would be made to explain it
further.
*
4 . 4. 2. 4 Correlation between and r^ with the help of
segregation equations
In the previous section r^ and r^ were correlated
by linear regression analysis. It should be recognized that it is
a purely empirical approach and does not have any segregation
model as basis. Now# attempts would be made to see how
segregation-models can be utilized for this purpose. It may be
noted that the nature of data, both in literature as well as in
the present investigation, is characterized by scatter. Hence,
the objective would be to attempt gross comparisons only. This is
the only rational approach according to the author.
It may also be pointed out here that the segregation
equations (Eqs.4.1 - 4.4 , in section 4.2) are applicable to
simple situations such as plane front solidification or at best
for dendritic solidification as encountered during columnar
growth of crystals. Hence, they are more applicable at
columnar-eguiaxed transition boundary rather than at the
centreline of the billet. However, a selective judicious
application to centreline is not ruled out.
232
(A) Equilibrium solidification model^®
The relevant equations are:
■'o = =s/=L
and ^
. . . (4.1)
(4.22)
me overall mass balance at the solidification front jives
=s *s =0 *0
combining above equations:
•♦.(4.23)
’'o *s + Cr (1 -
.. (4.24)
and
r •
i + *0 (Ito - 1)
(4.25)
Therefore, segregation ratios of carbon and sulphur would be:
“•*•1 - fg (K° - 1) ...(4.25a)
c
and I 1 - fg (k® - 1) ...(4.25b)
s
At any location under consideration f„ is same for both carbon
s
and sulphur.
kf - 1
—2 • b = a constant ...(4.26)
k" - 1
From Table 4.1, the value of b-0.82. For CET boundary, values of
r - l] have been plotted against l] for various samples
4.18. A line with elope b equal to 0.82 is also shown. It
234
is clear that there is no agreement of experimental data with
prediction of eguation (4.26). This is not surprising because the
equilibrium solidification model is hardly applicable to
continuously cast steel billets where solidification is fairly
fast.
(B) Schell’s equation and modified Schell’s equation^^’^^
Schell’s equation and modified Schell’s equation are
described in section 4.2 (viz., Eqs.(4.2) and (4.11)). From
Schell’s equation, segregation ratio of solute element can be
expressed as:
r
(1 - f^)
. . . (4.2)
Whereas, the modified Schell’s equation gives the following
equation of segregation ratio:
Cl - *3)
...(4.11)
From Schell’s equation the segregation ratio of carbon and
sulphur can be expressed as follows:
inr^ =
1
0 0
1)
ln(l -
... (4.27)
lnr„ =
s
-
1)
ln(l - fg)
... (4.28)
Therefore, at a fixed value of fg common to both carbon and
sulphur (i.e. at a particular location in the billet)
235
In Tg _ ^
“ m = a constant ...( 4 . 29 )
0
It IS evident from Eqs.(4.2) and (4.li) that the Scheil's
and modified Scheil's equations are of identical forms. Hence,
modified Scheil's equation would also give the same kind of
correlation as in Eq.(4.29).
From the values of and k= (Table 4.1) the value of
constant m (Eq.(4.29)) turns out to be 1.225. In Fig.4.i9 the
ratios of Inr^/lnr^ have been plotted against the corresponding
billet sample numbers for the GET boundary as well as the
centreline. The averages of Inr^/lnr^ data corresponding to GET
and centreline are presented in the figure as solid lines. The
line corresponding to Scheil's or modified Scheil's equation is
also shown in Fig. 4. 19. The apparent value of constant m (i.e.
m') in Eq.4.29 for the horizontal line is 1.235 for GET boundary.
It may be noted that this matches fairly well with the value of m
= 1.225 obtained from Eq. (4.29). On the other hand, m' = 0.72 for
the centreline segregation, and it does not match at all with
prediction of Eq.(4.29)
The agreement between m' corresponding to GET boundary with
the prediction based on Scheil' s/modified Scheil's equation has
been attributed to the following factors:
(i) As mentioned in Sec. 4.2, unlike equilibriiim
solidification model, Scheil's or modified Scheil's equation does
not assume uniformity of composition in the solid phase (i.e. no
diffusion in solid) . Also, the influence of fluid flow has been
taken into account in these models up to some extent (e.g.
inre/lnr,
236
Sample number
Fig. 4.19: inrs/Inrc values of billet samples for the
centreline and CET boundary.
237
modified Sohdil's equation). Hence, these models are fairly
closer to the real situation than the equilibrium solidification
model.
(ii) By considering the ratio of In r /In r , influence of
S C
fluid flow and other common factors get eliminated due to
cancellation effect.
(iix) As mentioned already in the beginning of this section/
that the Scheil's equation has better applicability in the
columnar dendritic region than in the equiaxed region. In the
columnar region solidification is predominantly unidirectional.
Therefore, the CET boundary can be roughly assumed to be a plane
front. On the other hand, the central equiaxed region has a more
complex solidification characteristics. Therefore, simple
segregation models would not be applicable to the centreline
segregation in CC products. However, over a small segregated
region such as spot segregation, Scheil's equation has been
applied even in the equiaxed zone in some of the previous
studies. Saeki et. al.^° have adopted the approach similar to the
present investigation and applied Schell's equation in the
analysis of seai-macrosegregation spots in slab. The
Investigators have reported a reasonable agreement between model
prediction and experimental data of P and Mn segregation in the
semi*-macrosegregation spots in slab.
However, the agreement with prediction based on Scheil s
equation (i.e. Eg. (4.29)) at the CET boundary in the present
study does not mean that these equations are fully applicable.
This will be further discussed in the next subsection.
238
(C) Equations using the concept of k
eff
Examples of these equations are that of Burton et al^^
(Eq-(4-3)) as well as by Takahashi et al^® (Eq.4.12). As
discussed in section 4.2, these two equations are based on
entirely different models. Any way both of them yield composition
variation with progress of solidification. As noted in Eq.4.4,
jCgff depends not only on but on other kinetic parameters as
well- These equations would lead to a relationship between r and
r as follows:
c
lnr„
s
Irir^
...(4.30)
However, it was not possible to assign values of k^^^. Hence, it
was not possible to test the applicability of Eg. (4.30) to the
segregation data of the present investigation.
4.4.2.S Relationship of r^ and r^ at CET boundary with
fractional solidification (fg)
As already stated, the values of fg at the CET
boundaries of different samples were calculated from the area
fraction of equiaxed zone (Table 4.4). Values of fg have been
presented in Table 4.7. As may be noted from Sec. 4.2 that the
segregation equations gave relationship between ( ' ^ °
and fg. For examining relationship between r and fg
decided to try only ScheiPs or modified Scheil's equation, since
only then, could be satisfectorlly employed for correlation
between r. and et the CET boundary (Pig.
s ^
239
As described earlier, from Scheil^s model, the following
correlations between r^, r^ and f^ is obtained.
Inr =
s
O'.
- 1) In (1 - f^)
. . . (4.31)
and
lnr„ =
c
O':
“ 1) In (1 - f )
s
...(4.32)
From
the modified Scheil^s
equation, the correlations
obtained
are as follows:
Inr^ =
- l)/e [ln(l - fg)j
... (4.33)
and
Inr^ *
c
- 1)/C [in (1 - fg)j
... (4.34)
Fig. 4. 20 presents Inr^ vs. In(l-fg) for all data points at
the CET boundary, and the same for Inr^ is presented in Fig. 4. 21.
Equations (4. 31) -(4. 34) show that with increasing In(l-fg) (i.e.
decreasing f_) both r and r should decrease. But the best fit
lines in Figs. 4.20 and 4.21 have negative slope (i.e. with
increasing and r^ actually decreased). This finding runs
counter to predictions of any of the segregation models mentioned
above. However, they are in agreement with segregation data in
transverse sections of continuously cast products as observed by
several investigators®^
A region of negative segregation around the high centreli
positive segregation zone has been reported y
investigators®"*®^ Take at al®^ have reported decreasing carbon
concentration between fg-0.64 to fg“0.85 in a 0.8 p
steel billet (110 m square). They observed this consis y
billets with both high as well as low superheat (27 C an
mrs
242
in another study, Miyazawa and Schwerdtfeger^^ observed a region
of negative segregation around the centreline concentration peak
at close to fg=0.95 (Fig. 4.1). In the present study varied
between 0.58 to 0.97. Therefore, from that point of view the
present finding is consistent with that reported in literature.
In a controlled laboratory experiment involving
solidification of liguid steel against a rotating water-cooled
copper chill, Takahashi et al observed that the concentration
of liquid and solid phases remained unchanged regardless of the
position. However, with rotation (i.e. fluid flow) solute
concentration in solid phase decreased with position away from
the chill, whereas, the liquid concentration increased
(Fig. 4. 22). The investigators attributed this to the washing
effect due to bulk liquid flow. However, there is no clear cut
explanation available in literature on the formation of zone of
59
low or negative segregation. Miyazawa and Schwerdtfeger
attributed the flow of heated fluid with lower solute
concentration in the mushy zone due to bulging to be the cause of
negative segregation around the centreline positive
macrosegregation. Moore^® attributed this to the fluctuating
nature of segregation profile. Settling of free crystals may also
be one of the causes.
243
D.stance from cMI (cm)
fv^ on the rate
r hulk liqoiCl flow I
FI,. «.22! SnfW«"“ “ concentration profile®
of solidlfice’-K’" (»)
1 fit ft 6 ^
of different sol laBoratory
1^6 in a control lea
Takahashl et a .
expat
244
4.5 CORRELATION OF MACROSEGREGATION AND MORPHOLOGY DATA WITH
PREDICTION OF HEAT TRANSFER MODEL
AS Stated in Chapter i, section 1.3 that one of the
objectives of the overall investigation was to try to correlate
predictions based on mathematical modelling of heat transfer with
the experimental observations on macrosegregation and morphology.
Such an attempt was made for the columnar-equiaxed transition
boundary. The procedure for this is outlined below;
1. The chemical composition at the GET boundary of a
billet section was employed to estimate the liguidus temperature
at that location with the help of Eq. (4.15). Table 4.8
presents the estimated values of for all samples.
2. Measurements on a macroetched billet section allowed
determination of the average distance of the CET boundary from
the centre of the section. This is being designated as d^, whose
values for the samples are reported in Table 4.8.
3. From the computer program for conjugate fluid flow-heat
transfer model, as presented in Chapter 3, temperatures were
calculated at various grid points corresponding to the casting
condition for the sample (Table 4.3).
4 . From the computer print out of the above temperature
field, the grid point corresponding to T^ was found out. This
yielded the calculated value of the distance of the CET boundary
from the centre of the billet section. This is being designated
as d^ .
6. The value ot d, as detenninad from the mathematical
aodel would depend on the pouring temperature of liquid steel. As
discussed in Section 4.3, attempts were made to experim n y
245
.easure temperature of liquid steel in the cc .ould or for the
pouring stream from the tundish to the mould. However, these
attempts failed because of experimental difficulties. Therefore,
the temperatures of liquid steel as measured by Immersion
thermocouple in the tundish are available only.
6. Hence, in order to arrive at the correct pouring
temperature, attempt was made to estimate the temperature loss
from tundish to mould. Table 4.8 presents two values of d , viz.
^®^^®sponds to uncorrected pouring temperature
and ^^2 to corrected tundish temperature as pouring
temperature. Values of d^^ and d^^ have been reported in Table
4.8. Sub section 4.5.1 discusses about determination of the
temperature loss from tundish to mould in continuous casting of
steel .
4,5.1 Estimation of Correct Pouring Temperature of
Liquid Steel
Again,
Correct pouring = Temperature measured in tundish
temperature - Loss of temperature from
tundish--to~mould (AT^^^)
... (4.35)
AT.
Lo«t
AT.
+ AT.
Loss, W
... (4.36)
in which AT « Loss of temperature due to pouring into
mould
and AT - Loss of temperature in the tundish up to the
tundish nozzle
246
^\os»,s (Eq.(4.36)) was calculated according to the
procedure outlined by Ghoeh^S. primarily due to
radiation from the surface of the teeming stream. From the plant
data estimated as 0.35°C. This is not significant,
and hence can be ignored. ^ is much larger than AT
* LoS8,S
In the literature few investigations could be located®®'^®®.
On the basis of measurements in the plant, Nemoto^^
determined mould superheat with varying tundish superheat (Fig.
4.23) .It shows that the AT^^^^ which is the difference between
tundish and mould superheat, ranges from approximately 10 - 30°C.
Robertson and Perkins^°° carried out an extensive
investigation on temperature loss of liquid steel in ladle and in
tundish. Their study consisted of measurements in plant,
mathematical modelling as well as water modelling on tundish. The
purpose of water model work of tundish was to essentially find
out the residence time distribution for each nozzle separately.
One of the tundishes simulated was the centre-filled billet
caster tundish of Temple borough plant of British Steel
Corporation^®'®. This was fitted with six nozzles for a six strand
CC machine. This was similar both in shape and geometry to the
six strand tundish at Tata Steel.
Robertson and Perkins^®® also carried out heat loss studies
in the proto- type tundish in plant. Residence times were
estimated from their water model data, in order to attempt an
estimate of AT for liquid metal at Tata Steel, further
LoM.W _
information were obtained from the investigators th g
correspondence .
SuperiiMt of Molien Stool m MoW
I (*C)
Fig. 4.23: Relationship between superheat of
molten steel in the tundish and
99
that in the mold .
248
Robertson and Perkins““ proposed the following correlation
for the average loss of temperature in tundish (i.e AT )
' * * LOSS,W^
AT == T
LOSS,W L
^r tQs \ + Qh W m c
.. . (4.37)
Where is the mean residence time of tundish. Q and Q are the
average heat flux densities through the surface of liquid steel
and through the walls respectively in the tundish. A and A are
H S
the surface areas of wall and bottom of tundish, and surface area
of the melt surface respectively. M is tundish capacity and c is
the specific heat of steel.
At the Tata Steel, residence time measurements had been
carried out in tundish water model and data have been reported
elsewhere^® Relevant data for the estimation of average heat
flux densities, Q and Q , were obtained from Robertson and
s w
Perkins^®®. For the Tata Steel tundish, actual residence times
were estimated from the residence times measured on water model
by multiplying the latter by the scaling factor using the
following expression^^^:
Scaling Factor = v~o~ j
m '^p
Where V is the tundish capacity and Q is the liquid flow rate,
and m and p refer to model and plant respectively.
For Tata Steel tundish, VpSl.S m^, Qp = 220 1 min » \ ®
and Q - 61.43 1 min'^ Using these data the estimated
M . ,
residence time was found to be 41 sec. for strand 4 of the bille
caster.
249
Using = 41 sec, a sample calculation based on Eq. (4.37)
gave a value of 8 °C drop in temperature in the tundish for
strand 4 , which matches with the temperature loss reported by
Robertson and Perkins for a similar tundish. However, the mean
residence time data obtained from Perkins through correspondence
was 3.5 min (=210 sec). Calculations based on this residence time
yielded a fairly higher tundish temperature loss. Considering all
these, finally it was decided to take AT = 10 °C in all the
LOSS
subsequent calculations.
4.5.2 Comparison of Measured Location of CET Botmdaries
with Those Predicted from Mathematical Model
Table 4.8 presents the computed values of the
distance of the CET boundary from centre of the billet section.
Pouring temperature i.e temperature of liquid steel entering the
mould, were either uncorrected (i.e. same as temperature of
liquid steel in the tundish) or by taking = 10 C as
discussed in the previous sub-section.
Fig. 4.24 shows a plot of d^^ vs. d^, where d^^ is the
predicted value of the distance of the CET boundary from centre
of the billet section when the pouring temperature was
uncorrected, d^ represents the same measured experimentally from
macrostructures. As the figure shows that the values of d^^ are
somewhat higher than d^. Table 4.8 presents the values P
deviation of d from d . It ranges from 1.5 to 30 pet
Jmi: m J •
Fig. 4. 25 shows the plot of d^^ X 2
calculated value of the CET boundary from the model
250
Table 4.8; Results of correlation between measured GET boundary
and those predicted by the mathematical model
s
A
M
P
L
E
T
L . CET
It
It
(""c)
A1
1509-4
A2
1510.2
B1
1511.8
B2
1513.3
Cl
1522.4
C2
1524.0
D1
1520.5
D2
1520-3
El
1508.4
E2
1509,0
FI
1517.2
F2
1517.5
G1
1519.0
G2
1518.7
HI
1524.9
H2
1524.3
I
1512,0
J1
1515.0
J2
1514.3
K1
1515.2
K2
1514.9
Distance of GET boundary from centre of
the billet (mm)
computed with tundish temp.
uncorrected
xt
corrected
X2
Deviation
(pet.)
^X2~^y
1.4
2.9
5.1
0.6
6.0
0.6
5.9
0.6
29.1
5.5
17.6
4.0
29.6
4.8
7.0
2.8
1.7
2.0
3.8
2.2
2.7
0.9
1.9
0.2
17.8
3.6
11.5
1.0
6.7
1,8
2.6
-4.7
16.7
-0.5
8.5
2.0
9.7
6.6
10.7
1.8
28.0
33.0
33.0
36.0
11.0
15.0
12.5
18.0
30.0
36.4
33.1
40.5
14.0
20.0
16.5
23.4
21.0
15.3
21.6
27.2
28.0
28.4
34.8
35.1
38.0
14.2
18.0
16.2
19.3
30.5
37.8
34.0
41.3
16.5
22.3
17.6
24.0
24.5
16 . 6
23.7
29.0
31.0
27.2
33.3
32.9
36.1
10.4
15.9
13.1
18.5
30.6
37.2
33.4
40.6
14.5
20.2
16.8
22.3
20.9
15.6
21.9
27.8
28.5
251
Rg.
4.24; Comparison between
CET boundory from
uncorrected pouring
measured and computed positions of
the centre of billet samples with
temperature.
252
Fig.
4 . 25 :
Comparison between measured
CET boundary from the centre
corrected pouring temperature.
and computed positions
of billet samples with
of
253
pouring temperature. It is evident from the figure that values of
ffi^tches very well. This finding tends to confirm the
reliability of both macrostructural measurements as well as the
conjugate fluid flow-heat transfer model. Therefore/ it is
proposed that the same model may also be employed to predict the
equiaxed zone size in continuously cast products.
4.6 SUMMARY AND CONCLUSIONS
Samples of 125 mm square billets of low carbon steel and the
corresponding plant data were collected from strand no. 4 of the
continuous casting unit of Tata Steel, Jamshedpur. The samples
were cut to a convenient size in order to carry out physical and
chemical examinations of transverse sections of the billets.
Ground and cleaned surfaces of billets were examined without
etching as well as after macroetching by warm 1:1 hydrochloric
acid-water solution with added hydrogen peroxide.
All visible macrostructural features of the transverse
sections of the billets were traced on transparent papers. These
were subsequently employed for measurements of fractions
chill, columnar equiaxed zone as well as to ascertain th
positions of columnar-equiaxed transition (CET) boundaries.
subsequent to the macrostructural examination, carbon and
sulphur contents of steel at the CET boundary as well as
centre of each billet section were determined. Samples were
collected by drilling and analyzed in an automatic carb
determinator.
Area percent of equiaxed zone
/A ) for transverse sections
254
ranged frota 3 to 42 percent. Equiaxed zones were mostly
asymmetric around the centres of the billet sections.
Temperature of molten steel was measured in the tundish
twice during each cast. For determination of tundish superheat
(AT), liquidus temperatures (T^^) were estimated for all the
compositions by correlations of with steel composition,
proposed by several workers. On the basis of this exercise, the
correlation of Thomas et al was accepted. AT ranged from 7 to
50 C. was found to decrease with increase in AT in agreement
with literature reports and the equation of the best fit line
was:
A„„ = 21.45
Eq
0.17 AT
Degree of segregation of carbon and sulphur (r^ and r_
c s
respectively) were calculated from analyses of drillings and from
compositions of liquid steel as provided by the plant.
Statistically speaking, r^ and r^ at the billet centres were
higher than those at corresponding GET boundaries, as expected.
Both for the centre and the GET boundary, r^ was linearly
correlated with r by the least square method. The best fit lines
were compared with those available in literature.
Segregation equations based on various models were tested
for their applicability at GET boundary. Equilibrium
solidification model did not agree with experimental data.
Predictions based on Scheil's equation or modified Scheil
equation yielded the relationship:
Inr^
s ^
Inr^
Theoretical value
of m is 1.225,
constant
and at GET boundary
255
experimental data yielded an average value ot 1.235 demonstrating
good agreement. No agreement was found with the data for centre
line.
The variations of and at CET boundary with fractional
solidification for the billet samples did not a^ree with
predictions of above equations. However they were qualitatively
consistent with observations reported in literature, and may be
attributed to complex movements of liquid and free crystals in
the solidifying pool of liquid steel.
Correlation of macrosegregation and morphological studies
with heat transfer modelling was one of the objectives of the
entire investigation reported here, and was attempted as follows.
Experiiaental values of location of the CET boundaries were
obtained from the macrostructures of billet sections. Conjugate
fluid flow-heat transfer model was employed to calculate
temperatures at various grid points. Location of CET boundaries
were found from these corresponding to a liquidus temperatures at
CET boundaries. These were carried out for both uncorrected and
corrected pouring temperatures.
It has been demonstrated that predicted values of location
of CET boundaries from the fluid flow heat transfer model agreed
closely with the experimental values for corrected pouring
temperatures of liquid steel.
The above agreement is being taken as an additional
confirmation of the reliability of the conjugate fluid flow-heat
transfer model. Therefore, It is proposed that the model may also
be employed to estimate equiaxed zone size in continue y
sections.
CHAPTER 5
SUMMARY AND CONCLUSIONS
I
The present study is concerned with continuous casting of
steel, and consists of the following:
(i) mathematical modelling of heat transfer by the
artificial effective thermal conductivity model
(ii) mathematical modelling by conjugate fluid flow - heat
transfer model
(iii) macrosegregation and morphology study in continuously
cast low carbon steel billets.
These have been presented in Chapters 2, 3, and 4
respectively. Each of these chapters has its own summary and
conclusions. This chapter presents the same of the entire study
in a consolidated fashion. It is a somewhat abridged version of
the same from the above chapters.
5.1 MATHEMATICAL MODELLING BY ARTIFICIAL EFFECTIVE THERMA
CONDUCTIVITY APPROACH
(i) Based on the concept of artificial effective thermal
conductivity approach, a steady state 3D heat flow model of
continuous casting of steel was developed, control volume based
finite difference procedure has been employed for the
solution of the governing heat flow equation. A ge
program, which incorporates Tri Diagonal Matrix Algorithm
solution of discretisation equation, has been deve ope
257
FORTRAN 77. The program is so written that computations in
eartesian as well as in cylindrical polar coordinate systems can
be performed in both 2-D and 3-D.
(ii) The present study has revealed that the axial
conduction term has a minor role to play so far as the modelling
of overall heat flow in CC is concerned.' Numerical solution
required the choice of grid configurations so as to make it grid
independent. Procedures applied to model heat flow in the mushy
zone as well as the surface boundary condition in the mould were
also found to affect the predictions somewhat.
(iii) In order to select a proper value and test its
sensitivity to computed results, values of were varied over
a wide range. Finally, it was decided to take = 7K for
further computation.
(iv) Model predictions have been assessed against three sets
of experimental data from literature for round and sc[uare billet
casters. However, in most of the cases the overall agreement
between predictions and experimental measurements of shell
thickness were not found to be satisfactory.
(V) It appears that the concept of artificial effective
thermal conductivity, as applied to the liquid pool to account
for the effect of fluid convection and turbulence on heat
transfer, is not adequate enough to describe various thermal
phenomena in continuous casting of steel realistically.
258
5.2 Mathematical Modelling by Conjugate Fluid Flow - Heat
Transfer Approach
(i) A steady state, two dimensional mathematical model
based on the concept of conjugate fluid flow and heat transfer
has been developed for continuous casting of steel.
(ii) Two-dimensional turbulent Navier Stokes equation has
been considered for the simulation of fluid flow in the liquid
pool and furthermore, a thermal buoyancy force term has been
incorporated in the axial direction momentum balance equation to
take into account the natural convection phenomena taking place
in the liquid pool of the solidifying casting.
(iii) The turbulence properties in the system was estimated
via the Pun - Spalding formula, based on which the average
effective viscosity was computed. Similarly, in the mushy zone,
resistance to the flow produced by the solid matrix has been
taken into account by increasing the viscosity to 20 times the
molecular viscosity of liquid steel.
(iv) In conjunction with these considerations, an
appropriate energy balance equation was considered, in which the
latent heat of solidification was estimated from the solid
fractions in the mushy zone assuming equilibrium solidification
of steel.
(V) The TEACH-T computer code, with considerable
modifications, was used for the numerical solution of the
governing fluid flow and heat transfer equations and thus,
deduce flow field, temperature field and pool profile in
continuously cast billets.
(Vi) prior to carrying out any coapariaon with experinental
259
measurements, influence of various approximations applied to the
mathematical model were analyzed computationally. Towards this,
the predicted shell thickness was found to be almost insensitive
to the precise value of effective viscosity. This in turn
revealed that the exact modelling of turbulence in the pool is
relatively less critical than has been originally anticipated.
However, modelling of flow in the mushy zone was found to have
some bearing on the predicted shell thickness, particularly in
the lower pool region. Similarly, influence of buoyancy induced
natural convection on the overall shell growth was found to be
almost insignificant.
(vii) Velocity and temperature fields were calculated for
three different CC sections. The predictions of velocity field
revealed that the flow of liquid steel was predominantly in the
axial direction for most of the central regions. Whereas, near
the solidification front some reverse flow were seen.
Furthermore, reverse flow was found to be significant only up to
few meters below the meniscus.
(viii) Comparison between predicted shell thickness and
corresponding experimental measurements reported in literature
indicated reasonable agreement between the two. Similarly,
comparison between the predictions of conjugate fluid flow and
heat transfer model and those derived via the artificial
effective thermal conductivity model demonstrated the superiority
of the former over the latter.
260
5.3 STUDY ON KACROSEGREGATION AND MORPHOLOGY
(i) Samples of 125 mm square billets of low carbon steel
and the corresponding plant data were collected from strand no. 4
of the continuous casting unit of Tata Steel, Jamshedpur. The
samples were cut to a convenient size in order to carry out
physical and chemical examinations of transverse sections of the
billets. Ground and cleaned surfaces of billets were examined
without etching as well as after macroetching by warm 1:1
hydrochloric acid-water solution with added hydrogen peroxide.
(ii) Area percent of equiaxed zone (A ) for transverse
Eq
sections ranged from 3 to 42 pet.. Equiaxed zones were mostly
asymmetric around the centres of the billet sections. Temperature
of molten steel was measured in the tundish twice during each
cast. Tundish superheat (AT) ranged from 7 to 50 °C. A^ was
found to decrease with increase in AT in agreement with
literature reports.
(iil) Subsequent to the macrostructural examination, carbon
and sulphur contents of steel at the columnar-equiaxed transition
boundary (GET) as well as at the centre of each billet section
were determined. Samples were collected by drilling and analyzed
in an automatic carbon-sulphur determinator.
(iv) Degree of segregation of carbon and sulphur (r^ and r^
respectively) were calculated from analyses of drillings and from
compositions of liquid steel as provided by the plant.
Statistically speaking, r^, and at the billet centres were
higher than those at corresponding GET boundaries, as expected.
Both for the centre and the GET boundary, r^ was linear y
correlated with r^ by the least square method. The best fit lines
261
w6irB co3Jip^ir0cl with thoso available in litaratur^
(V) segregation equations based on various models were
tested for their applicability at CRT boundary. Equilibrium
solidification model did not agree with experimental data.
Predictions based on Soheil's equation or modified Schell's
equation yielded the relationship;
Inr^
a = a constant
Theoretical value of m is 1.225, and at CET boundary
axparimental data yielded an average value of 1.235 demonstrating
good agreement. No agreement was found with the data for centre
line.
(vi) The variations of r^ and r^ at CET boundary with
fractional solidification (f ) for the billet samples did not
agree with predictions of above equations. However they were
qualitatively consistent with observations reported in
literature, and may be attributed to complex movements of liquid
and free crystals in the solidifying pool of liquid steel.
5.4 CORRELATION OF MACROSEGREGATION AND MORPHOLOGICAL STUDIES
WITH HEAT TRANSFER MODELLING
(i) This was one of the the objectives of the entire
investigation reported here, and was attempted as follows.
Experimental values of location of the CET boundaries were
obtained from the macrostructures of billet sections, conjugate
fluid flow-heat transfer model was employed to calculate
262
temperatures at various grid points. Location of GET boundaries
were found from these corresponding to liquidus temperatures at
CET boundaries. These were carried out for both \ancorrected and
corrected pouring temperatures.
(ii) It has been demonstrated that predicted values of
locations of CET boundaries from the fluid flow heat transfer
model agreed closely with the experimental values for corrected
pouring temperatures of liquid steel.
(iii) The above agreement is being taken as an additional
confirmation of the reliability of the conjugate fluid flow-heat
transfer model.
(iv) It is proposed that the model may also be employed to
estimate equiaxed zone size.
5.5 SUGGESTIONS FOR FURTHER WORK
(i) In the present conjugate fluid flow - heat transfe
model, a somewhat simplified modelling approaches have been
adopted to take into account the turbulence phenomena
liquid pool, for the reasons already described in Chapter 3.
Therefore, it would be desirable to carry out
comparative study between the present approach and that by
application of standard turbulence model.
(ii) Improved treatments for fluid and heat
4. A iry literature. Inclusion of some of
mushy zone has been reported in , - „
iA Ho desirable. Similarly, for
these treatments in the model would
o nd fraction in the mushy zone,
the estimation of solid fraction
^ lo are available which if
nonequiUbrlum aoUdifioation mo e
included In the model would make the con
263
transfer model further closer to the real situations.
(iii) in the present study, blockage ratio/oell porosity
method have been adopted to take into account the influence of
solidified shell on the velocity field and to assign the
prescribed velocity (i.e. the casting speed) in the completely
solidified region. Alternative approach to this is the high
viscosity method. A comparative study between these techniques
from the view point of convergence and the accuracy of prediction
is desirable.
(iv) Extension of the present conjugate fluid flow-heat
transfer model to three dimension is desirable. Also, in the
present study pouring of liquid steel through straight nozzle has
been considered. However, in industries, particularly in slab
casting submerged entry nozzles are commonly employed. Therefore,
the present study should be extended to the submerged pouring
condition also.
(v) Mathematical modelling in conjunction with in-plant
measurements is desirable.
(vi) A mathematical model involving coupled fluid flow, heat
transfer, as well as mass transfer phenomena can provide the
complete description of continuous casting process.
(vii) The key findings in macrosegregation and morphological
studies at the CET boundary, viz. some agreement with Schell's
equations, as well as agreement with heat transfer model ought to
be established by carrying out more experiments. Measurement of
temperature of liquid steel in mould or teeming stream should be
an integral part of such study.
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